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Theorem fin23lem25 8743
Description: Lemma for fin23 8808. 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 3547 . . . . . . . 8  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 php3 7755 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  A  ~<  B )
3 sdomnen 7596 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
42, 3syl 17 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  C.  B )  ->  -.  A  ~~  B )
54ex 435 . . . . . . . 8  |-  ( B  e.  Fin  ->  ( A  C.  B  ->  -.  A  ~~  B ) )
61, 5syl5bir 221 . . . . . . 7  |-  ( B  e.  Fin  ->  (
( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
76adantl 467 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
87expd 437 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  C_  B  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
9 dfpss2 3547 . . . . . . . . 9  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  B  =  A ) )
10 eqcom 2429 . . . . . . . . . . 11  |-  ( B  =  A  <->  A  =  B )
1110notbii 297 . . . . . . . . . 10  |-  ( -.  B  =  A  <->  -.  A  =  B )
1211anbi2i 698 . . . . . . . . 9  |-  ( ( B  C_  A  /\  -.  B  =  A
)  <->  ( B  C_  A  /\  -.  A  =  B ) )
139, 12bitri 252 . . . . . . . 8  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  =  B ) )
14 php3 7755 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
15 sdomnen 7596 . . . . . . . . . . 11  |-  ( B 
~<  A  ->  -.  B  ~~  A )
16 ensym 7616 . . . . . . . . . . 11  |-  ( A 
~~  B  ->  B  ~~  A )
1715, 16nsyl 124 . . . . . . . . . 10  |-  ( B 
~<  A  ->  -.  A  ~~  B )
1814, 17syl 17 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  -.  A  ~~  B )
1918ex 435 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
2013, 19syl5bir 221 . . . . . . 7  |-  ( A  e.  Fin  ->  (
( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2120adantr 466 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  C_  A  /\  -.  A  =  B )  ->  -.  A  ~~  B ) )
2221expd 437 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  C_  A  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
238, 22jaod 381 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( A  C_  B  \/  B  C_  A
)  ->  ( -.  A  =  B  ->  -.  A  ~~  B ) ) )
24233impia 1202 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  =  B  ->  -.  A  ~~  B ) )
2524con4d 108 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~~  B  ->  A  =  B ) )
26 eqeng 7601 . . 3  |-  ( A  e.  Fin  ->  ( A  =  B  ->  A 
~~  B ) )
27263ad2ant1 1026 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  =  B  ->  A  ~~  B
) )
2825, 27impbid 193 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433    C. wpss 3434   class class class wbr 4417    ~~ cen 7565    ~< csdm 7567   Fincfn 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572
This theorem is referenced by:  fin23lem23  8745  fin1a2lem9  8827
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