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Theorem fiuneneq 30986
Description: Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Assertion
Ref Expression
fiuneneq  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  <->  A  =  B ) )

Proof of Theorem fiuneneq
StepHypRef Expression
1 simp2 997 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  e.  Fin )
2 enfi 7737 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
323ad2ant1 1017 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
41, 3mpbid 210 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  e.  Fin )
5 unfi 7788 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
61, 4, 5syl2anc 661 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  e.  Fin )
7 ssun1 3667 . . . . . 6  |-  A  C_  ( A  u.  B
)
87a1i 11 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  C_  ( A  u.  B
) )
9 simp3 998 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  A )
109ensymd 7567 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  ( A  u.  B
) )
11 fisseneq 7732 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  A  C_  ( A  u.  B )  /\  A  ~~  ( A  u.  B
) )  ->  A  =  ( A  u.  B ) )
126, 8, 10, 11syl3anc 1228 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  ( A  u.  B ) )
13 ssun2 3668 . . . . . 6  |-  B  C_  ( A  u.  B
)
1413a1i 11 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  C_  ( A  u.  B
) )
15 simp1 996 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  B )
16 entr 7568 . . . . . . 7  |-  ( ( ( A  u.  B
)  ~~  A  /\  A  ~~  B )  -> 
( A  u.  B
)  ~~  B )
179, 15, 16syl2anc 661 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  B )
1817ensymd 7567 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  ~~  ( A  u.  B
) )
19 fisseneq 7732 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  B  C_  ( A  u.  B )  /\  B  ~~  ( A  u.  B
) )  ->  B  =  ( A  u.  B ) )
206, 14, 18, 19syl3anc 1228 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  =  ( A  u.  B ) )
2112, 20eqtr4d 2511 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  B )
22213expia 1198 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  ->  A  =  B ) )
23 enrefg 7548 . . . 4  |-  ( A  e.  Fin  ->  A  ~~  A )
2423adantl 466 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  A  ~~  A )
25 unidm 3647 . . . . 5  |-  ( A  u.  A )  =  A
26 uneq2 3652 . . . . 5  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
2725, 26syl5eqr 2522 . . . 4  |-  ( A  =  B  ->  A  =  ( A  u.  B ) )
2827breq1d 4457 . . 3  |-  ( A  =  B  ->  ( A  ~~  A  <->  ( A  u.  B )  ~~  A
) )
2924, 28syl5ibcom 220 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( A  =  B  ->  ( A  u.  B )  ~~  A
) )
3022, 29impbid 191 1  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   class class class wbr 4447    ~~ cen 7514   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-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  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-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  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-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521
This theorem is referenced by:  idomsubgmo  30987
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