Theorem djuen 7425

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 6914	. . . . . . . 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 7295	. . . . . 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 6957	. . . . 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 7295	. . . . . 6 |- ( B e. _V -> (inl " B ) ~~ B )
10	2, 9	simpl2im 386	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " B ) ~~ B )
11	10	ensymd 6956	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> B ~~ (inl " B ) )
12		entr 6957	. . . 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 6914	. . . . . . . 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 7296	. . . . . 6 |- ( C e. _V -> (inr " C ) ~~ C )
18	16, 17	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ C )
19		entr 6957	. . . . 5 |- ( ( (inr " C ) ~~ C /\ C ~~ D ) -> (inr " C ) ~~ D )
20	18, 19	sylancom 420	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ D )
21		eninr 7296	. . . . . 6 |- ( D e. _V -> (inr " D ) ~~ D )
22	15, 21	simpl2im 386	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " D ) ~~ D )
23	22	ensymd 6956	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> D ~~ (inr " D ) )
24		entr 6957	. . . 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 7262	. . . 4 |- ( (inl " A ) i^i (inr " C ) ) = (/)
27	26	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) i^i (inr " C ) ) = (/) )
28		djuin 7262	. . . 4 |- ( (inl " B ) i^i (inr " D ) ) = (/)
29	28	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " B ) i^i (inr " D ) ) = (/) )
30		unen 6990	. . 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 1274	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) u. (inr " C ) ) ~~ ( (inl " B ) u. (inr " D ) ) )
32		djuun 7265	. 2 |- ( (inl " A ) u. (inr " C ) ) = ( A ⊔ C )
33		djuun 7265	. 2 |- ( (inl " B ) u. (inr " D ) ) = ( B ⊔ D )
34	31, 32, 33	3brtr3g 4121	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   i^i cin 3199   (/)c0 3494   class class class wbr 4088   "cima 4728   ~~ cen 6906   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-er 6701  df-en 6909  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  djuenun  7426  exmidunben  13046  enctlem  13052