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Theorem fin23lem25 8206
Description: Lemma for fin23 8271. In a chain of finite sets, equinumerousity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
fin23lem25  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  <->  A  =  B ) )

Proof of Theorem fin23lem25
StepHypRef Expression
1 dfpss2 3434 . . . . . . . 8  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 php3 7295 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  A  ~<  B )
3 sdomnen 7138 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
42, 3syl 16 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  -.  A  ~~  B )
54ex 425 . . . . . . . 8  |-  ( B  e.  Fin  ->  ( A  C.  B  ->  -.  A  ~~  B ) )
61, 5syl5bir 211 . . . . . . 7  |-  ( B  e.  Fin  ->  (
( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
76adantl 454 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
87exp3a 427 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  C_  B  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
9 dfpss2 3434 . . . . . . . . 9  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  B  =  A ) )
10 eqcom 2440 . . . . . . . . . . 11  |-  ( B  =  A  <->  A  =  B )
1110notbii 289 . . . . . . . . . 10  |-  ( -.  B  =  A  <->  -.  A  =  B )
1211anbi2i 677 . . . . . . . . 9  |-  ( ( B  C_  A  /\  -.  B  =  A
)  <->  ( B  C_  A  /\  -.  A  =  B ) )
139, 12bitri 242 . . . . . . . 8  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  =  B ) )
14 php3 7295 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
15 sdomnen 7138 . . . . . . . . . . 11  |-  ( B 
~<  A  ->  -.  B  ~~  A )
16 ensym 7158 . . . . . . . . . . 11  |-  ( A 
~~  B  ->  B  ~~  A )
1715, 16nsyl 116 . . . . . . . . . 10  |-  ( B 
~<  A  ->  -.  A  ~~  B )
1814, 17syl 16 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  -.  A  ~~  B )
1918ex 425 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
2013, 19syl5bir 211 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2120adantr 453 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2221exp3a 427 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  C_  A  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
238, 22jaod 371 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  \/  B  C_  A
)  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
24233impia 1151 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  =  B  ->  -.  A  ~~  B ) )
2524con4d 100 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  ->  A  =  B ) )
26 eqeng 7143 . . 3  |-  ( A  e.  Fin  ->  ( A  =  B  ->  A 
~~  B ) )
27263ad2ant1 979 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  =  B  ->  A  ~~  B
) )
2825, 27impbid 185 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322    C. wpss 3323   class class class wbr 4214    ~~ cen 7108    ~< csdm 7110   Fincfn 7111
This theorem is referenced by:  fin23lem23  8208  fin1a2lem9  8290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115
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