Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fiuneneq Unicode version

Theorem fiuneneq 27513
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 7079 . . . . . . . 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 7124 . . . . . 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 3338 . . . . . 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 6910 . . . . . 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 7074 . . . . 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 3339 . . . . . 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 6913 . . . . . . 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 6910 . . . . . 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 7074 . . . . 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 2318 . . 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 6893 . . . 4  |-  ( A  e.  Fin  ->  A  ~~  A )
2625adantl 452 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  A  ~~  A )
27 unidm 3318 . . . . 5  |-  ( A  u.  A )  =  A
28 uneq2 3323 . . . . 5  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
2927, 28syl5eqr 2329 . . . 4  |-  ( A  =  B  ->  A  =  ( A  u.  B ) )
3029breq1d 4033 . . 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 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   class class class wbr 4023    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  idomsubgmo  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
  Copyright terms: Public domain W3C validator