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Theorem infpssrlem5 8678
Description: Lemma for infpssr 8679. (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 8676 . . 3  |-  ( ph  ->  G : om --> A )
6 simpll 751 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ph )
7 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  c  e.  om )
8 simpr 459 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  b  e.  c )
91, 2, 3, 4infpssrlem4 8677 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  om 
/\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b )
)
106, 7, 8, 9syl3anc 1226 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b
) )
1110necomd 2725 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  b )  =/=  ( G `  c
) )
12 simpll 751 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ph )
13 simplrl 759 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  b  e.  om )
14 simpr 459 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  c  e.  b )
151, 2, 3, 4infpssrlem4 8677 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  om 
/\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c )
)
1612, 13, 14, 15syl3anc 1226 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) )
1711, 16jaodan 783 . . . . . . 7  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  ( b  e.  c  \/  c  e.  b ) )  -> 
( G `  b
)  =/=  ( G `
 c ) )
1817ex 432 . . . . . 6  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( b  e.  c  \/  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) ) )
1918necon2bd 2669 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  -.  ( b  e.  c  \/  c  e.  b ) ) )
20 nnord 6681 . . . . . . 7  |-  ( b  e.  om  ->  Ord  b )
21 nnord 6681 . . . . . . 7  |-  ( c  e.  om  ->  Ord  c )
22 ordtri3 4903 . . . . . . 7  |-  ( ( Ord  b  /\  Ord  c )  ->  (
b  =  c  <->  -.  (
b  e.  c  \/  c  e.  b ) ) )
2320, 21, 22syl2an 475 . . . . . 6  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2423adantl 464 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2519, 24sylibrd 234 . . . 4  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  b  =  c ) )
2625ralrimivva 2875 . . 3  |-  ( ph  ->  A. b  e.  om  A. c  e.  om  (
( G `  b
)  =  ( G `
 c )  -> 
b  =  c ) )
27 dff13 6141 . . 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 662 . 2  |-  ( ph  ->  G : om -1-1-> A
)
29 f1domg 7528 . 2  |-  ( A  e.  V  ->  ( G : om -1-1-> A  ->  om 
~<_  A ) )
3028, 29syl5com 30 1  |-  ( ph  ->  ( A  e.  V  ->  om  ~<_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458    C_ wss 3461   class class class wbr 4439   Ord word 4866   `'ccnv 4987    |` cres 4990   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ` cfv 5570   omcom 6673   reccrdg 7067    ~<_ cdom 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-dom 7511
This theorem is referenced by:  infpssr  8679
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