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Theorem fiuneneq 35774
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 1006 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  e.  Fin )
2 enfi 7785 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
323ad2ant1 1026 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
41, 3mpbid 213 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  e.  Fin )
5 unfi 7835 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
61, 4, 5syl2anc 665 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  e.  Fin )
7 ssun1 3626 . . . . . 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 1007 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  A )
109ensymd 7618 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  ( A  u.  B
) )
11 fisseneq 7780 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  A  C_  ( A  u.  B )  /\  A  ~~  ( A  u.  B
) )  ->  A  =  ( A  u.  B ) )
126, 8, 10, 11syl3anc 1264 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  ( A  u.  B ) )
13 ssun2 3627 . . . . . 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 1005 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  ~~  B )
16 entr 7619 . . . . . . 7  |-  ( ( ( A  u.  B
)  ~~  A  /\  A  ~~  B )  -> 
( A  u.  B
)  ~~  B )
179, 15, 16syl2anc 665 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  ( A  u.  B )  ~~  B )
1817ensymd 7618 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  ~~  ( A  u.  B
) )
19 fisseneq 7780 . . . . 5  |-  ( ( ( A  u.  B
)  e.  Fin  /\  B  C_  ( A  u.  B )  /\  B  ~~  ( A  u.  B
) )  ->  B  =  ( A  u.  B ) )
206, 14, 18, 19syl3anc 1264 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  B  =  ( A  u.  B ) )
2112, 20eqtr4d 2464 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin  /\  ( A  u.  B )  ~~  A )  ->  A  =  B )
22213expia 1207 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  ->  A  =  B ) )
23 enrefg 7599 . . . 4  |-  ( A  e.  Fin  ->  A  ~~  A )
2423adantl 467 . . 3  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  A  ~~  A )
25 unidm 3606 . . . . 5  |-  ( A  u.  A )  =  A
26 uneq2 3611 . . . . 5  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
2725, 26syl5eqr 2475 . . . 4  |-  ( A  =  B  ->  A  =  ( A  u.  B ) )
2827breq1d 4427 . . 3  |-  ( A  =  B  ->  ( A  ~~  A  <->  ( A  u.  B )  ~~  A
) )
2924, 28syl5ibcom 223 . 2  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( A  =  B  ->  ( A  u.  B )  ~~  A
) )
3022, 29impbid 193 1  |-  ( ( A  ~~  B  /\  A  e.  Fin )  ->  ( ( A  u.  B )  ~~  A  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    u. cun 3431    C_ wss 3433   class class class wbr 4417    ~~ cen 7565   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-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  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-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  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-pred 5390  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572
This theorem is referenced by:  idomsubgmo  35775
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