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Theorem djuen 7072
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 6640 . . . . . . . 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 6982 . . . . . 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 6678 . . . . 5  |-  ( ( (inl " A ) 
~~  A  /\  A  ~~  B )  ->  (inl " A )  ~~  B
)
85, 6, 7syl2anc 408 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " A )  ~~  B )
9 eninl 6982 . . . . . 6  |-  ( B  e.  _V  ->  (inl " B )  ~~  B
)
102, 9simpl2im 383 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " B )  ~~  B )
1110ensymd 6677 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  B  ~~  (inl " B
) )
12 entr 6678 . . . 4  |-  ( ( (inl " A ) 
~~  B  /\  B  ~~  (inl " B ) )  ->  (inl " A
)  ~~  (inl " B
) )
138, 11, 12syl2anc 408 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inl " A )  ~~  (inl " B ) )
14 encv 6640 . . . . . . . 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 6983 . . . . . 6  |-  ( C  e.  _V  ->  (inr " C )  ~~  C
)
1816, 17syl 14 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  C )
19 entr 6678 . . . . 5  |-  ( ( (inr " C ) 
~~  C  /\  C  ~~  D )  ->  (inr " C )  ~~  D
)
2018, 19sylancom 416 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  D )
21 eninr 6983 . . . . . 6  |-  ( D  e.  _V  ->  (inr " D )  ~~  D
)
2215, 21simpl2im 383 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " D )  ~~  D )
2322ensymd 6677 . . . 4  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  D  ~~  (inr " D
) )
24 entr 6678 . . . 4  |-  ( ( (inr " C ) 
~~  D  /\  D  ~~  (inr " D ) )  ->  (inr " C
)  ~~  (inr " D
) )
2520, 23, 24syl2anc 408 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
(inr " C )  ~~  (inr " D ) )
26 djuin 6949 . . . 4  |-  ( (inl " A )  i^i  (inr " C ) )  =  (/)
2726a1i 9 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " A
)  i^i  (inr " C
) )  =  (/) )
28 djuin 6949 . . . 4  |-  ( (inl " B )  i^i  (inr " D ) )  =  (/)
2928a1i 9 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " B
)  i^i  (inr " D
) )  =  (/) )
30 unen 6710 . . 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 1217 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( (inl " A
)  u.  (inr " C ) )  ~~  ( (inl " B )  u.  (inr " D
) ) )
32 djuun 6952 . 2  |-  ( (inl " A )  u.  (inr " C ) )  =  ( A C )
33 djuun 6952 . 2  |-  ( (inl " B )  u.  (inr " D ) )  =  ( B D )
3431, 32, 333brtr3g 3961 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A C )  ~~  ( B D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686    u. cun 3069    i^i cin 3070   (/)c0 3363   class class class wbr 3929   "cima 4542    ~~ cen 6632   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-er 6429  df-en 6635  df-dju 6923  df-inl 6932  df-inr 6933
This theorem is referenced by:  djuenun  7073  exmidunben  11950  enctlem  11956
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