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Theorem fiuneneq 27616
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 956 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  e.  Fin )
2 enfi 7095 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
323ad2ant1 976 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
41, 3mpbid 201 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  e.  Fin )
5 unfi 7140 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
61, 4, 5syl2anc 642 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  e.  Fin )
7 ssun1 3351 . . . . . 6  |-  A  C_  ( A  u.  B
)
87a1i 10 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  C_  ( A  u.  B
) )
9 simp3 957 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  A )
10 ensym 6926 . . . . . 6  |-  ( ( A  u.  B ) 
~~  A  ->  A  ~~  ( A  u.  B
) )
119, 10syl 15 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  ( A  u.  B
) )
12 fisseneq 7090 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  A  C_  ( A  u.  B )  /\  A  ~~  ( A  u.  B
) )  ->  A  =  ( A  u.  B ) )
136, 8, 11, 12syl3anc 1182 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  ( A  u.  B ) )
14 ssun2 3352 . . . . . 6  |-  B  C_  ( A  u.  B
)
1514a1i 10 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  C_  ( A  u.  B
) )
16 simp1 955 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  B )
17 entr 6929 . . . . . . 7  |-  ( ( ( A  u.  B
)  ~~  A  /\  A  ~~  B )  -> 
( A  u.  B
)  ~~  B )
189, 16, 17syl2anc 642 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  B )
19 ensym 6926 . . . . . 6  |-  ( ( A  u.  B ) 
~~  B  ->  B  ~~  ( A  u.  B
) )
2018, 19syl 15 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  ~~  ( A  u.  B
) )
21 fisseneq 7090 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  B  C_  ( A  u.  B )  /\  B  ~~  ( A  u.  B
) )  ->  B  =  ( A  u.  B ) )
226, 15, 20, 21syl3anc 1182 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  =  ( A  u.  B ) )
2313, 22eqtr4d 2331 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  B )
24233expia 1153 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  ->  A  =  B ) )
25 enrefg 6909 . . . 4  |-  ( A  e.  Fin  ->  A  ~~  A )
2625adantl 452 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  A  ~~  A )
27 unidm 3331 . . . . 5  |-  ( A  u.  A )  =  A
28 uneq2 3336 . . . . 5  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
2927, 28syl5eqr 2342 . . . 4  |-  ( A  =  B  ->  A  =  ( A  u.  B ) )
3029breq1d 4049 . . 3  |-  ( A  =  B  ->  ( A  ~~  A  <->  ( A  u.  B )  ~~  A
) )
3126, 30syl5ibcom 211 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( A  =  B  ->  ( A  u.  B )  ~~  A
) )
3224, 31impbid 183 1  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   class class class wbr 4039    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  idomsubgmo  27617
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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