MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oawordeulem Unicode version

Theorem oawordeulem 6636
Description: Lemma for oawordex 6639. (Contributed by NM, 11-Dec-2004.)
Hypotheses
Ref Expression
oawordeulem.1  |-  A  e.  On
oawordeulem.2  |-  B  e.  On
oawordeulem.3  |-  S  =  { y  e.  On  |  B  C_  ( A  +o  y ) }
Assertion
Ref Expression
oawordeulem  |-  ( A 
C_  B  ->  E! x  e.  On  ( A  +o  x )  =  B )
Distinct variable groups:    x, y, A    x, B, y    x, S
Allowed substitution hint:    S( y)

Proof of Theorem oawordeulem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 oawordeulem.3 . . . . . 6  |-  S  =  { y  e.  On  |  B  C_  ( A  +o  y ) }
2 ssrab2 3334 . . . . . 6  |-  { y  e.  On  |  B  C_  ( A  +o  y
) }  C_  On
31, 2eqsstri 3284 . . . . 5  |-  S  C_  On
4 oawordeulem.2 . . . . . . 7  |-  B  e.  On
5 oawordeulem.1 . . . . . . . 8  |-  A  e.  On
6 oaword2 6635 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
74, 5, 6mp2an 653 . . . . . . 7  |-  B  C_  ( A  +o  B
)
8 oveq2 5950 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  +o  y )  =  ( A  +o  B
) )
98sseq2d 3282 . . . . . . . 8  |-  ( y  =  B  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  B ) ) )
109, 1elrab2 3001 . . . . . . 7  |-  ( B  e.  S  <->  ( B  e.  On  /\  B  C_  ( A  +o  B
) ) )
114, 7, 10mpbir2an 886 . . . . . 6  |-  B  e.  S
12 ne0i 3537 . . . . . 6  |-  ( B  e.  S  ->  S  =/=  (/) )
1311, 12ax-mp 8 . . . . 5  |-  S  =/=  (/)
14 oninton 4670 . . . . 5  |-  ( ( S  C_  On  /\  S  =/=  (/) )  ->  |^| S  e.  On )
153, 13, 14mp2an 653 . . . 4  |-  |^| S  e.  On
16 onzsl 4716 . . . . . . . 8  |-  ( |^| S  e.  On  <->  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) ) )
1715, 16mpbi 199 . . . . . . 7  |-  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) )
18 oveq2 5950 . . . . . . . . . . 11  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  ( A  +o  (/) ) )
19 oa0 6599 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
205, 19ax-mp 8 . . . . . . . . . . 11  |-  ( A  +o  (/) )  =  A
2118, 20syl6eq 2406 . . . . . . . . . 10  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  A )
2221sseq1d 3281 . . . . . . . . 9  |-  ( |^| S  =  (/)  ->  (
( A  +o  |^| S )  C_  B  <->  A 
C_  B ) )
2322biimprd 214 . . . . . . . 8  |-  ( |^| S  =  (/)  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
24 oveq2 5950 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  -> 
( A  +o  |^| S )  =  ( A  +o  suc  z
) )
25 oasuc 6607 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
265, 25mpan 651 . . . . . . . . . . . 12  |-  ( z  e.  On  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z ) )
2724, 26sylan9eqr 2412 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  =  suc  ( A  +o  z ) )
28 vex 2867 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
2928sucid 4550 . . . . . . . . . . . . . 14  |-  z  e. 
suc  z
30 eleq2 2419 . . . . . . . . . . . . . 14  |-  ( |^| S  =  suc  z  -> 
( z  e.  |^| S 
<->  z  e.  suc  z
) )
3129, 30mpbiri 224 . . . . . . . . . . . . 13  |-  ( |^| S  =  suc  z  -> 
z  e.  |^| S
)
3215oneli 4579 . . . . . . . . . . . . . 14  |-  ( z  e.  |^| S  ->  z  e.  On )
331inteqi 3945 . . . . . . . . . . . . . . . . 17  |-  |^| S  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }
3433eleq2i 2422 . . . . . . . . . . . . . . . 16  |-  ( z  e.  |^| S  <->  z  e.  |^|
{ y  e.  On  |  B  C_  ( A  +o  y ) } )
35 oveq2 5950 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  ( A  +o  y )  =  ( A  +o  z
) )
3635sseq2d 3282 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  z ) ) )
3736onnminsb 4674 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  (
z  e.  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  -.  B  C_  ( A  +o  z ) ) )
3834, 37syl5bi 208 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  -.  B  C_  ( A  +o  z ) ) )
39 oacl 6618 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  z
)  e.  On )
405, 39mpan 651 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  On  ->  ( A  +o  z )  e.  On )
41 ontri1 4505 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  On  /\  ( A  +o  z
)  e.  On )  ->  ( B  C_  ( A  +o  z
)  <->  -.  ( A  +o  z )  e.  B
) )
424, 40, 41sylancr 644 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  ( B  C_  ( A  +o  z )  <->  -.  ( A  +o  z )  e.  B ) )
4342con2bid 319 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
( A  +o  z
)  e.  B  <->  -.  B  C_  ( A  +o  z
) ) )
4438, 43sylibrd 225 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  ( A  +o  z
)  e.  B ) )
4532, 44mpcom 32 . . . . . . . . . . . . 13  |-  ( z  e.  |^| S  ->  ( A  +o  z )  e.  B )
464onordi 4576 . . . . . . . . . . . . . 14  |-  Ord  B
47 ordsucss 4688 . . . . . . . . . . . . . 14  |-  ( Ord 
B  ->  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z )  C_  B ) )
4846, 47ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z
)  C_  B )
4931, 45, 483syl 18 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  ->  suc  ( A  +o  z
)  C_  B )
5049adantl 452 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  suc  ( A  +o  z )  C_  B )
5127, 50eqsstrd 3288 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  C_  B )
5251rexlimiva 2738 . . . . . . . . 9  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  +o  |^| S )  C_  B
)
5352a1d 22 . . . . . . . 8  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
54 iunss 4022 . . . . . . . . . . 11  |-  ( U_ z  e.  |^| S ( A  +o  z ) 
C_  B  <->  A. z  e.  |^| S ( A  +o  z )  C_  B )
554onelssi 4580 . . . . . . . . . . . 12  |-  ( ( A  +o  z )  e.  B  ->  ( A  +o  z )  C_  B )
5645, 55syl 15 . . . . . . . . . . 11  |-  ( z  e.  |^| S  ->  ( A  +o  z )  C_  B )
5754, 56mprgbir 2689 . . . . . . . . . 10  |-  U_ z  e.  |^| S ( A  +o  z )  C_  B
58 oalim 6615 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  ( |^| S  e.  _V  /\ 
Lim  |^| S ) )  ->  ( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
595, 58mpan 651 . . . . . . . . . . 11  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
6059sseq1d 3281 . . . . . . . . . 10  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( ( A  +o  |^| S )  C_  B  <->  U_ z  e.  |^| S
( A  +o  z
)  C_  B )
)
6157, 60mpbiri 224 . . . . . . . . 9  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  C_  B
)
6261a1d 22 . . . . . . . 8  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
6323, 53, 623jaoi 1245 . . . . . . 7  |-  ( (
|^| S  =  (/)  \/ 
E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e. 
_V  /\  Lim  |^| S
) )  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
6417, 63ax-mp 8 . . . . . 6  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  C_  B )
659rspcev 2960 . . . . . . . . 9  |-  ( ( B  e.  On  /\  B  C_  ( A  +o  B ) )  ->  E. y  e.  On  B  C_  ( A  +o  y ) )
664, 7, 65mp2an 653 . . . . . . . 8  |-  E. y  e.  On  B  C_  ( A  +o  y )
67 nfcv 2494 . . . . . . . . . 10  |-  F/_ y B
68 nfcv 2494 . . . . . . . . . . 11  |-  F/_ y A
69 nfcv 2494 . . . . . . . . . . 11  |-  F/_ y  +o
70 nfrab1 2796 . . . . . . . . . . . 12  |-  F/_ y { y  e.  On  |  B  C_  ( A  +o  y ) }
7170nfint 3951 . . . . . . . . . . 11  |-  F/_ y |^| { y  e.  On  |  B  C_  ( A  +o  y ) }
7268, 69, 71nfov 5965 . . . . . . . . . 10  |-  F/_ y
( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7367, 72nfss 3249 . . . . . . . . 9  |-  F/ y  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
74 oveq2 5950 . . . . . . . . . 10  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( A  +o  y )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } ) )
7574sseq2d 3282 . . . . . . . . 9  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) ) )
7673, 75onminsb 4669 . . . . . . . 8  |-  ( E. y  e.  On  B  C_  ( A  +o  y
)  ->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) )
7766, 76ax-mp 8 . . . . . . 7  |-  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
7833oveq2i 5953 . . . . . . 7  |-  ( A  +o  |^| S )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7977, 78sseqtr4i 3287 . . . . . 6  |-  B  C_  ( A  +o  |^| S
)
8064, 79jctir 524 . . . . 5  |-  ( A 
C_  B  ->  (
( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
81 eqss 3270 . . . . 5  |-  ( ( A  +o  |^| S
)  =  B  <->  ( ( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
8280, 81sylibr 203 . . . 4  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  =  B )
83 oveq2 5950 . . . . . 6  |-  ( x  =  |^| S  -> 
( A  +o  x
)  =  ( A  +o  |^| S ) )
8483eqeq1d 2366 . . . . 5  |-  ( x  =  |^| S  -> 
( ( A  +o  x )  =  B  <-> 
( A  +o  |^| S )  =  B ) )
8584rspcev 2960 . . . 4  |-  ( (
|^| S  e.  On  /\  ( A  +o  |^| S )  =  B )  ->  E. x  e.  On  ( A  +o  x )  =  B )
8615, 82, 85sylancr 644 . . 3  |-  ( A 
C_  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
87 eqtr3 2377 . . . . 5  |-  ( ( ( A  +o  x
)  =  B  /\  ( A  +o  y
)  =  B )  ->  ( A  +o  x )  =  ( A  +o  y ) )
88 oacan 6630 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  On  /\  y  e.  On )  ->  (
( A  +o  x
)  =  ( A  +o  y )  <->  x  =  y ) )
895, 88mp3an1 1264 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( A  +o  x )  =  ( A  +o  y )  <-> 
x  =  y ) )
9087, 89syl5ib 210 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) )
9190rgen2a 2685 . . 3  |-  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y )
9286, 91jctir 524 . 2  |-  ( A 
C_  B  ->  ( E. x  e.  On  ( A  +o  x
)  =  B  /\  A. x  e.  On  A. y  e.  On  (
( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) ) )
93 oveq2 5950 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
9493eqeq1d 2366 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  =  B  <->  ( A  +o  y )  =  B ) )
9594reu4 3035 . 2  |-  ( E! x  e.  On  ( A  +o  x )  =  B  <->  ( E. x  e.  On  ( A  +o  x )  =  B  /\  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) ) )
9692, 95sylibr 203 1  |-  ( A 
C_  B  ->  E! x  e.  On  ( A  +o  x )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   E!wreu 2621   {crab 2623   _Vcvv 2864    C_ wss 3228   (/)c0 3531   |^|cint 3941   U_ciun 3984   Ord word 4470   Oncon0 4471   Lim wlim 4472   suc csuc 4473  (class class class)co 5942    +o coa 6560
This theorem is referenced by:  oawordeu  6637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-oadd 6567
  Copyright terms: Public domain W3C validator