Theorem djuen 7212

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

Step	Hyp	Ref	Expression
1		encv 6748	. . . . . . . 8 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
2	1	adantr 276	. . . . . . 7 |- ( ( A ~~ B /\ C ~~ D ) -> ( A e. _V /\ B e. _V ) )
3	2	simpld 112	. . . . . 6 |- ( ( A ~~ B /\ C ~~ D ) -> A e. _V )
4		eninl 7098	. . . . . 6 |- ( A e. _V -> (inl " A ) ~~ A )
5	3, 4	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ A )
6		simpl 109	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> A ~~ B )
7		entr 6786	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ B ) -> (inl " A ) ~~ B )
8	5, 6, 7	syl2anc 411	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ B )
9		eninl 7098	. . . . . 6 |- ( B e. _V -> (inl " B ) ~~ B )
10	2, 9	simpl2im 386	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " B ) ~~ B )
11	10	ensymd 6785	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> B ~~ (inl " B ) )
12		entr 6786	. . . 4 |- ( ( (inl " A ) ~~ B /\ B ~~ (inl " B ) ) -> (inl " A ) ~~ (inl " B ) )
13	8, 11, 12	syl2anc 411	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ (inl " B ) )
14		encv 6748	. . . . . . . 8 |- ( C ~~ D -> ( C e. _V /\ D e. _V ) )
15	14	adantl 277	. . . . . . 7 |- ( ( A ~~ B /\ C ~~ D ) -> ( C e. _V /\ D e. _V ) )
16	15	simpld 112	. . . . . 6 |- ( ( A ~~ B /\ C ~~ D ) -> C e. _V )
17		eninr 7099	. . . . . 6 |- ( C e. _V -> (inr " C ) ~~ C )
18	16, 17	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ C )
19		entr 6786	. . . . 5 |- ( ( (inr " C ) ~~ C /\ C ~~ D ) -> (inr " C ) ~~ D )
20	18, 19	sylancom 420	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ D )
21		eninr 7099	. . . . . 6 |- ( D e. _V -> (inr " D ) ~~ D )
22	15, 21	simpl2im 386	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " D ) ~~ D )
23	22	ensymd 6785	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> D ~~ (inr " D ) )
24		entr 6786	. . . 4 |- ( ( (inr " C ) ~~ D /\ D ~~ (inr " D ) ) -> (inr " C ) ~~ (inr " D ) )
25	20, 23, 24	syl2anc 411	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ (inr " D ) )
26		djuin 7065	. . . 4 |- ( (inl " A ) i^i (inr " C ) ) = (/)
27	26	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) i^i (inr " C ) ) = (/) )
28		djuin 7065	. . . 4 |- ( (inl " B ) i^i (inr " D ) ) = (/)
29	28	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " B ) i^i (inr " D ) ) = (/) )
30		unen 6818	. . 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 ) ) )
31	13, 25, 27, 29, 30	syl22anc 1239	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) u. (inr " C ) ) ~~ ( (inl " B ) u. (inr " D ) ) )
32		djuun 7068	. 2 |- ( (inl " A ) u. (inr " C ) ) = ( A ⊔ C )
33		djuun 7068	. 2 |- ( (inl " B ) u. (inr " D ) ) = ( B ⊔ D )
34	31, 32, 33	3brtr3g 4038	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   u. cun 3129   i^i cin 3130   (/)c0 3424   class class class wbr 4005   "cima 4631   ~~ cen 6740   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-er 6537  df-en 6743  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  djuenun  7213  exmidunben  12429  enctlem  12435