MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infpssrlem5 Structured version   Unicode version

Theorem infpssrlem5 8187
Description: Lemma for infpssr 8188. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Hypotheses
Ref Expression
infpssrlem.a  |-  ( ph  ->  B  C_  A )
infpssrlem.c  |-  ( ph  ->  F : B -1-1-onto-> A )
infpssrlem.d  |-  ( ph  ->  C  e.  ( A 
\  B ) )
infpssrlem.e  |-  G  =  ( rec ( `' F ,  C )  |`  om )
Assertion
Ref Expression
infpssrlem5  |-  ( ph  ->  ( A  e.  V  ->  om  ~<_  A ) )

Proof of Theorem infpssrlem5
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infpssrlem.a . . . 4  |-  ( ph  ->  B  C_  A )
2 infpssrlem.c . . . 4  |-  ( ph  ->  F : B -1-1-onto-> A )
3 infpssrlem.d . . . 4  |-  ( ph  ->  C  e.  ( A 
\  B ) )
4 infpssrlem.e . . . 4  |-  G  =  ( rec ( `' F ,  C )  |`  om )
51, 2, 3, 4infpssrlem3 8185 . . 3  |-  ( ph  ->  G : om --> A )
6 simpll 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ph )
7 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  c  e.  om )
8 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  b  e.  c )
91, 2, 3, 4infpssrlem4 8186 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  om 
/\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b )
)
106, 7, 8, 9syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b
) )
1110necomd 2687 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  b )  =/=  ( G `  c
) )
12 simpll 731 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ph )
13 simplrl 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  b  e.  om )
14 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  c  e.  b )
151, 2, 3, 4infpssrlem4 8186 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  om 
/\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c )
)
1612, 13, 14, 15syl3anc 1184 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) )
1711, 16jaodan 761 . . . . . . 7  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  ( b  e.  c  \/  c  e.  b ) )  -> 
( G `  b
)  =/=  ( G `
 c ) )
1817ex 424 . . . . . 6  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( b  e.  c  \/  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) ) )
1918necon2bd 2653 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  -.  ( b  e.  c  \/  c  e.  b ) ) )
20 nnord 4853 . . . . . . 7  |-  ( b  e.  om  ->  Ord  b )
21 nnord 4853 . . . . . . 7  |-  ( c  e.  om  ->  Ord  c )
22 ordtri3 4617 . . . . . . 7  |-  ( ( Ord  b  /\  Ord  c )  ->  (
b  =  c  <->  -.  (
b  e.  c  \/  c  e.  b ) ) )
2320, 21, 22syl2an 464 . . . . . 6  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2423adantl 453 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2519, 24sylibrd 226 . . . 4  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  b  =  c ) )
2625ralrimivva 2798 . . 3  |-  ( ph  ->  A. b  e.  om  A. c  e.  om  (
( G `  b
)  =  ( G `
 c )  -> 
b  =  c ) )
27 dff13 6004 . . 3  |-  ( G : om -1-1-> A  <->  ( G : om --> A  /\  A. b  e.  om  A. c  e.  om  ( ( G `
 b )  =  ( G `  c
)  ->  b  =  c ) ) )
285, 26, 27sylanbrc 646 . 2  |-  ( ph  ->  G : om -1-1-> A
)
29 f1domg 7127 . 2  |-  ( A  e.  V  ->  ( G : om -1-1-> A  ->  om 
~<_  A ) )
3028, 29syl5com 28 1  |-  ( ph  ->  ( A  e.  V  ->  om  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    \ cdif 3317    C_ wss 3320   class class class wbr 4212   Ord word 4580   omcom 4845   `'ccnv 4877    |` cres 4880   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454   reccrdg 6667    ~<_ cdom 7107
This theorem is referenced by:  infpssr  8188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-dom 7111
  Copyright terms: Public domain W3C validator