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Theorem fin23lem25 8705
Description: Lemma for fin23 8770. In a chain of finite sets, equinumerosity 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 3589 . . . . . . . 8  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 php3 7704 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  A  ~<  B )
3 sdomnen 7545 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
42, 3syl 16 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  -.  A  ~~  B )
54ex 434 . . . . . . . 8  |-  ( B  e.  Fin  ->  ( A  C.  B  ->  -.  A  ~~  B ) )
61, 5syl5bir 218 . . . . . . 7  |-  ( B  e.  Fin  ->  (
( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
76adantl 466 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
87expd 436 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  C_  B  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
9 dfpss2 3589 . . . . . . . . 9  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  B  =  A ) )
10 eqcom 2476 . . . . . . . . . . 11  |-  ( B  =  A  <->  A  =  B )
1110notbii 296 . . . . . . . . . 10  |-  ( -.  B  =  A  <->  -.  A  =  B )
1211anbi2i 694 . . . . . . . . 9  |-  ( ( B  C_  A  /\  -.  B  =  A
)  <->  ( B  C_  A  /\  -.  A  =  B ) )
139, 12bitri 249 . . . . . . . 8  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  =  B ) )
14 php3 7704 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
15 sdomnen 7545 . . . . . . . . . . 11  |-  ( B 
~<  A  ->  -.  B  ~~  A )
16 ensym 7565 . . . . . . . . . . 11  |-  ( A 
~~  B  ->  B  ~~  A )
1715, 16nsyl 121 . . . . . . . . . 10  |-  ( B 
~<  A  ->  -.  A  ~~  B )
1814, 17syl 16 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  -.  A  ~~  B )
1918ex 434 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
2013, 19syl5bir 218 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2120adantr 465 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2221expd 436 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  C_  A  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
238, 22jaod 380 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  \/  B  C_  A
)  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
24233impia 1193 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  =  B  ->  -.  A  ~~  B ) )
2524con4d 105 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  ->  A  =  B ) )
26 eqeng 7550 . . 3  |-  ( A  e.  Fin  ->  ( A  =  B  ->  A 
~~  B ) )
27263ad2ant1 1017 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  =  B  ->  A  ~~  B
) )
2825, 27impbid 191 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476    C. wpss 3477   class class class wbr 4447    ~~ cen 7514    ~< csdm 7516   Fincfn 7517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521
This theorem is referenced by:  fin23lem23  8707  fin1a2lem9  8789
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