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Theorem fiuneneq 29574
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 989 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  e.  Fin )
2 enfi 7541 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
323ad2ant1 1009 . . . . . . 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 7591 . . . . . 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 3531 . . . . . 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 990 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  A )
109ensymd 7372 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  ( A  u.  B
) )
11 fisseneq 7536 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  A  C_  ( A  u.  B )  /\  A  ~~  ( A  u.  B
) )  ->  A  =  ( A  u.  B ) )
126, 8, 10, 11syl3anc 1218 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  ( A  u.  B ) )
13 ssun2 3532 . . . . . 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 988 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  B )
16 entr 7373 . . . . . . 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 7372 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  ~~  ( A  u.  B
) )
19 fisseneq 7536 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  B  C_  ( A  u.  B )  /\  B  ~~  ( A  u.  B
) )  ->  B  =  ( A  u.  B ) )
206, 14, 18, 19syl3anc 1218 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  =  ( A  u.  B ) )
2112, 20eqtr4d 2478 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  B )
22213expia 1189 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  ->  A  =  B ) )
23 enrefg 7353 . . . 4  |-  ( A  e.  Fin  ->  A  ~~  A )
2423adantl 466 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  A  ~~  A )
25 unidm 3511 . . . . 5  |-  ( A  u.  A )  =  A
26 uneq2 3516 . . . . 5  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
2725, 26syl5eqr 2489 . . . 4  |-  ( A  =  B  ->  A  =  ( A  u.  B ) )
2827breq1d 4314 . . 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 965    = wceq 1369    e. wcel 1756    u. cun 3338    C_ wss 3340   class class class wbr 4304    ~~ cen 7319   Fincfn 7322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326
This theorem is referenced by:  idomsubgmo  29575
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