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Theorem infpssrlem5 8497
Description: Lemma for infpssr 8498. (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 8495 . . 3  |-  ( ph  ->  G : om --> A )
6 simpll 753 . . . . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  b  e.  c )
91, 2, 3, 4infpssrlem4 8496 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  om 
/\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b )
)
106, 7, 8, 9syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b
) )
1110necomd 2640 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  b )  =/=  ( G `  c
) )
12 simpll 753 . . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  c  e.  b )
151, 2, 3, 4infpssrlem4 8496 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  om 
/\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c )
)
1612, 13, 14, 15syl3anc 1218 . . . . . . . 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 434 . . . . . 6  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( b  e.  c  \/  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) ) )
1918necon2bd 2684 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  -.  ( b  e.  c  \/  c  e.  b ) ) )
20 nnord 6505 . . . . . . 7  |-  ( b  e.  om  ->  Ord  b )
21 nnord 6505 . . . . . . 7  |-  ( c  e.  om  ->  Ord  c )
22 ordtri3 4776 . . . . . . 7  |-  ( ( Ord  b  /\  Ord  c )  ->  (
b  =  c  <->  -.  (
b  e.  c  \/  c  e.  b ) ) )
2320, 21, 22syl2an 477 . . . . . 6  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2423adantl 466 . . . . 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 2829 . . 3  |-  ( ph  ->  A. b  e.  om  A. c  e.  om  (
( G `  b
)  =  ( G `
 c )  -> 
b  =  c ) )
27 dff13 5992 . . 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 664 . 2  |-  ( ph  ->  G : om -1-1-> A
)
29 f1domg 7350 . 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 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736    \ cdif 3346    C_ wss 3349   class class class wbr 4313   Ord word 4739   `'ccnv 4860    |` cres 4863   -->wf 5435   -1-1->wf1 5436   -1-1-onto->wf1o 5438   ` cfv 5439   omcom 6497   reccrdg 6886    ~<_ cdom 7329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-recs 6853  df-rdg 6887  df-dom 7333
This theorem is referenced by:  infpssr  8498
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