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Theorem djuen 7161
Description: Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
djuen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A C )  ~~  ( B D )
)

Proof of Theorem djuen
StepHypRef Expression
1 encv 6706 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
21adantr 274 . . . . . . 7  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  e.  _V  /\  B  e.  _V )
)
32simpld 111 . . . . . 6  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  A  e.  _V )
4 eninl 7056 . . . . . 6  |-  ( A  e.  _V  ->  (inl " A )  ~~  A
)
53, 4syl 14 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " A )  ~~  A )
6 simpl 108 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  A  ~~  B )
7 entr 6744 . . . . 5  |-  ( ( (inl " A ) 
~~  A  /\  A  ~~  B )  ->  (inl " A )  ~~  B
)
85, 6, 7syl2anc 409 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " A )  ~~  B )
9 eninl 7056 . . . . . 6  |-  ( B  e.  _V  ->  (inl " B )  ~~  B
)
102, 9simpl2im 384 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " B )  ~~  B )
1110ensymd 6743 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  B  ~~  (inl " B
) )
12 entr 6744 . . . 4  |-  ( ( (inl " A ) 
~~  B  /\  B  ~~  (inl " B ) )  ->  (inl " A
)  ~~  (inl " B
) )
138, 11, 12syl2anc 409 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " A )  ~~  (inl " B ) )
14 encv 6706 . . . . . . . 8  |-  ( C 
~~  D  ->  ( C  e.  _V  /\  D  e.  _V ) )
1514adantl 275 . . . . . . 7  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( C  e.  _V  /\  D  e.  _V )
)
1615simpld 111 . . . . . 6  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  C  e.  _V )
17 eninr 7057 . . . . . 6  |-  ( C  e.  _V  ->  (inr " C )  ~~  C
)
1816, 17syl 14 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  C )
19 entr 6744 . . . . 5  |-  ( ( (inr " C ) 
~~  C  /\  C  ~~  D )  ->  (inr " C )  ~~  D
)
2018, 19sylancom 417 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  D )
21 eninr 7057 . . . . . 6  |-  ( D  e.  _V  ->  (inr " D )  ~~  D
)
2215, 21simpl2im 384 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " D )  ~~  D )
2322ensymd 6743 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  D  ~~  (inr " D
) )
24 entr 6744 . . . 4  |-  ( ( (inr " C ) 
~~  D  /\  D  ~~  (inr " D ) )  ->  (inr " C
)  ~~  (inr " D
) )
2520, 23, 24syl2anc 409 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  (inr " D ) )
26 djuin 7023 . . . 4  |-  ( (inl " A )  i^i  (inr " C ) )  =  (/)
2726a1i 9 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " A
)  i^i  (inr " C
) )  =  (/) )
28 djuin 7023 . . . 4  |-  ( (inl " B )  i^i  (inr " D ) )  =  (/)
2928a1i 9 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " B
)  i^i  (inr " D
) )  =  (/) )
30 unen 6776 . . 3  |-  ( ( ( (inl " A
)  ~~  (inl " B
)  /\  (inr " C
)  ~~  (inr " D
) )  /\  (
( (inl " A
)  i^i  (inr " C
) )  =  (/)  /\  ( (inl " B
)  i^i  (inr " D
) )  =  (/) ) )  ->  (
(inl " A )  u.  (inr " C ) )  ~~  ( (inl " B )  u.  (inr " D ) ) )
3113, 25, 27, 29, 30syl22anc 1228 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " A
)  u.  (inr " C ) )  ~~  ( (inl " B )  u.  (inr " D
) ) )
32 djuun 7026 . 2  |-  ( (inl " A )  u.  (inr " C ) )  =  ( A C )
33 djuun 7026 . 2  |-  ( (inl " B )  u.  (inr " D ) )  =  ( B D )
3431, 32, 333brtr3g 4012 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A C )  ~~  ( B D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1342    e. wcel 2135   _Vcvv 2724    u. cun 3112    i^i cin 3113   (/)c0 3407   class class class wbr 3979   "cima 4604    ~~ cen 6698   ⊔ cdju 6996  inlcinl 7004  inrcinr 7005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-id 4268  df-iord 4341  df-on 4343  df-suc 4346  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-1st 6103  df-2nd 6104  df-1o 6378  df-er 6495  df-en 6701  df-dju 6997  df-inl 7006  df-inr 7007
This theorem is referenced by:  djuenun  7162  exmidunben  12353  enctlem  12359
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