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Theorem cantnflem2 7392
Description: Lemma for cantnf 7395. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
cantnf.1  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnf.2  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnf.3  |-  ( ph  -> 
(/)  e.  C )
Assertion
Ref Expression
cantnflem2  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Distinct variable groups:    x, w, y, z, B    w, C, x, y, z    w, A, x, y, z    x, S, y, z    ph, x, y, z
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)

Proof of Theorem cantnflem2
StepHypRef Expression
1 cantnfs.2 . . 3  |-  ( ph  ->  A  e.  On )
2 cantnfs.3 . . . . . . . . . 10  |-  ( ph  ->  B  e.  On )
3 oecl 6536 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 cantnf.1 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
6 onelon 4417 . . . . . . . . 9  |-  ( ( ( A  ^o  B
)  e.  On  /\  C  e.  ( A  ^o  B ) )  ->  C  e.  On )
74, 5, 6syl2anc 642 . . . . . . . 8  |-  ( ph  ->  C  e.  On )
8 cantnf.3 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  C )
9 ondif1 6500 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 645 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
11 eldifn 3299 . . . . . . 7  |-  ( C  e.  ( On  \  1o )  ->  -.  C  e.  1o )
1210, 11syl 15 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
13 ssel 3174 . . . . . . 7  |-  ( ( A  ^o  B ) 
C_  1o  ->  ( C  e.  ( A  ^o  B )  ->  C  e.  1o ) )
145, 13syl5com 26 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  B )  C_  1o  ->  C  e.  1o ) )
1512, 14mtod 168 . . . . 5  |-  ( ph  ->  -.  ( A  ^o  B )  C_  1o )
16 difss 3303 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
17 oe0m 6517 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
182, 17syl 15 . . . . . . . . 9  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
1918sseq1d 3205 . . . . . . . 8  |-  ( ph  ->  ( ( (/)  ^o  B
)  C_  1o  <->  ( 1o  \  B )  C_  1o ) )
2016, 19mpbiri 224 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
21 oveq1 5865 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2221sseq1d 3205 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2320, 22syl5ibrcom 213 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
24 oe1m 6543 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
25 eqimss 3230 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
262, 24, 253syl 18 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
27 oveq1 5865 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2827sseq1d 3205 . . . . . . 7  |-  ( A  =  1o  ->  (
( A  ^o  B
)  C_  1o  <->  ( 1o  ^o  B )  C_  1o ) )
2926, 28syl5ibrcom 213 . . . . . 6  |-  ( ph  ->  ( A  =  1o 
->  ( A  ^o  B
)  C_  1o )
)
3023, 29jaod 369 . . . . 5  |-  ( ph  ->  ( ( A  =  (/)  \/  A  =  1o )  ->  ( A  ^o  B )  C_  1o ) )
3115, 30mtod 168 . . . 4  |-  ( ph  ->  -.  ( A  =  (/)  \/  A  =  1o ) )
32 elpri 3660 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
33 df2o3 6492 . . . . 5  |-  2o  =  { (/) ,  1o }
3432, 33eleq2s 2375 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3531, 34nsyl 113 . . 3  |-  ( ph  ->  -.  A  e.  2o )
36 eldif 3162 . . 3  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
371, 35, 36sylanbrc 645 . 2  |-  ( ph  ->  A  e.  ( On 
\  2o ) )
3837, 10jca 518 1  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   (/)c0 3455   {cpr 3641   {copab 4076   Oncon0 4392   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^o coe 6478   CNF ccnf 7362
This theorem is referenced by:  cantnflem3  7393  cantnflem4  7394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485
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