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Theorem fin23lem25 7966
Description: Lemma for fin23 8031. 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 3274 . . . . . . . 8  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 php3 7063 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  A  ~<  B )
3 sdomnen 6906 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
42, 3syl 15 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  -.  A  ~~  B )
54ex 423 . . . . . . . 8  |-  ( B  e.  Fin  ->  ( A  C.  B  ->  -.  A  ~~  B ) )
61, 5syl5bir 209 . . . . . . 7  |-  ( B  e.  Fin  ->  (
( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
76adantl 452 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
87exp3a 425 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  C_  B  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
9 dfpss2 3274 . . . . . . . . 9  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  B  =  A ) )
10 eqcom 2298 . . . . . . . . . . 11  |-  ( B  =  A  <->  A  =  B )
1110notbii 287 . . . . . . . . . 10  |-  ( -.  B  =  A  <->  -.  A  =  B )
1211anbi2i 675 . . . . . . . . 9  |-  ( ( B  C_  A  /\  -.  B  =  A
)  <->  ( B  C_  A  /\  -.  A  =  B ) )
139, 12bitri 240 . . . . . . . 8  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  =  B ) )
14 php3 7063 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
15 sdomnen 6906 . . . . . . . . . . 11  |-  ( B 
~<  A  ->  -.  B  ~~  A )
16 ensym 6926 . . . . . . . . . . 11  |-  ( A 
~~  B  ->  B  ~~  A )
1715, 16nsyl 113 . . . . . . . . . 10  |-  ( B 
~<  A  ->  -.  A  ~~  B )
1814, 17syl 15 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  -.  A  ~~  B )
1918ex 423 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
2013, 19syl5bir 209 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2120adantr 451 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2221exp3a 425 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  C_  A  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
238, 22jaod 369 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  \/  B  C_  A
)  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
24233impia 1148 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  =  B  ->  -.  A  ~~  B ) )
2524con4d 97 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  ->  A  =  B ) )
26 eqeng 6911 . . 3  |-  ( A  e.  Fin  ->  ( A  =  B  ->  A 
~~  B ) )
27263ad2ant1 976 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  =  B  ->  A  ~~  B
) )
2825, 27impbid 183 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165    C. wpss 3166   class class class wbr 4039    ~~ cen 6876    ~< csdm 6878   Fincfn 6879
This theorem is referenced by:  fin23lem23  7968  fin1a2lem9  8050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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