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Theorem infpssrlem5 7933
Description: Lemma for infpssr 7934. (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 7931 . . 3  |-  ( ph  ->  G : om --> A )
6 simpll 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ph )
7 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  c  e.  om )
8 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  b  e.  c )
91, 2, 3, 4infpssrlem4 7932 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  om 
/\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b )
)
106, 7, 8, 9syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  c )  =/=  ( G `  b
) )
1110necomd 2529 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  b  e.  c )  ->  ( G `  b )  =/=  ( G `  c
) )
12 simpll 730 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ph )
13 simplrl 736 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  b  e.  om )
14 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  c  e.  b )
151, 2, 3, 4infpssrlem4 7932 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  om 
/\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c )
)
1612, 13, 14, 15syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) )
1711, 16jaodan 760 . . . . . . 7  |-  ( ( ( ph  /\  (
b  e.  om  /\  c  e.  om )
)  /\  ( b  e.  c  \/  c  e.  b ) )  -> 
( G `  b
)  =/=  ( G `
 c ) )
1817ex 423 . . . . . 6  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( b  e.  c  \/  c  e.  b )  ->  ( G `  b )  =/=  ( G `  c
) ) )
1918necon2bd 2495 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  -.  ( b  e.  c  \/  c  e.  b ) ) )
20 nnord 4664 . . . . . . 7  |-  ( b  e.  om  ->  Ord  b )
21 nnord 4664 . . . . . . 7  |-  ( c  e.  om  ->  Ord  c )
22 ordtri3 4428 . . . . . . 7  |-  ( ( Ord  b  /\  Ord  c )  ->  (
b  =  c  <->  -.  (
b  e.  c  \/  c  e.  b ) ) )
2320, 21, 22syl2an 463 . . . . . 6  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2423adantl 452 . . . . 5  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( b  =  c  <->  -.  ( b  e.  c  \/  c  e.  b ) ) )
2519, 24sylibrd 225 . . . 4  |-  ( (
ph  /\  ( b  e.  om  /\  c  e. 
om ) )  -> 
( ( G `  b )  =  ( G `  c )  ->  b  =  c ) )
2625ralrimivva 2635 . . 3  |-  ( ph  ->  A. b  e.  om  A. c  e.  om  (
( G `  b
)  =  ( G `
 c )  -> 
b  =  c ) )
27 dff13 5783 . . 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 645 . 2  |-  ( ph  ->  G : om -1-1-> A
)
29 f1domg 6881 . 2  |-  ( A  e.  V  ->  ( G : om -1-1-> A  ->  om 
~<_  A ) )
3028, 29syl5com 26 1  |-  ( ph  ->  ( A  e.  V  ->  om  ~<_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   class class class wbr 4023   Ord word 4391   omcom 4656   `'ccnv 4688    |` cres 4691   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255   reccrdg 6422    ~<_ cdom 6861
This theorem is referenced by:  infpssr  7934
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-recs 6388  df-rdg 6423  df-dom 6865
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