Theorem djuen 7567

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 7028	. . . . . . . 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 7437	. . . . . 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 7071	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ B ) -> (inl " A ) ~~ B )
8	5, 6, 7	syl2anc 415	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ B )
9		eninl 7437	. . . . . 6 |- ( B e. _V -> (inl " B ) ~~ B )
10	2, 9	simpl2im 390	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " B ) ~~ B )
11	10	ensymd 7070	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> B ~~ (inl " B ) )
12		entr 7071	. . . 4 |- ( ( (inl " A ) ~~ B /\ B ~~ (inl " B ) ) -> (inl " A ) ~~ (inl " B ) )
13	8, 11, 12	syl2anc 415	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ (inl " B ) )
14		encv 7028	. . . . . . . 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 7438	. . . . . 6 |- ( C e. _V -> (inr " C ) ~~ C )
18	16, 17	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ C )
19		entr 7071	. . . . 5 |- ( ( (inr " C ) ~~ C /\ C ~~ D ) -> (inr " C ) ~~ D )
20	18, 19	sylancom 424	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ D )
21		eninr 7438	. . . . . 6 |- ( D e. _V -> (inr " D ) ~~ D )
22	15, 21	simpl2im 390	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " D ) ~~ D )
23	22	ensymd 7070	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> D ~~ (inr " D ) )
24		entr 7071	. . . 4 |- ( ( (inr " C ) ~~ D /\ D ~~ (inr " D ) ) -> (inr " C ) ~~ (inr " D ) )
25	20, 23, 24	syl2anc 415	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ (inr " D ) )
26		djuin 7404	. . . 4 |- ( (inl " A ) i^i (inr " C ) ) = (/)
27	26	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) i^i (inr " C ) ) = (/) )
28		djuin 7404	. . . 4 |- ( (inl " B ) i^i (inr " D ) ) = (/)
29	28	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " B ) i^i (inr " D ) ) = (/) )
30		unen 7105	. . 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 1279	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) u. (inr " C ) ) ~~ ( (inl " B ) u. (inr " D ) ) )
32		djuun 7407	. 2 |- ( (inl " A ) u. (inr " C ) ) = ( A ⊔ C )
33		djuun 7407	. 2 |- ( (inl " B ) u. (inr " D ) ) = ( B ⊔ D )
34	31, 32, 33	3brtr3g 4163	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   class class class wbr 4130   "cima 4777   ~~ cen 7020   ⊔ cdju 7377  inlcinl 7385  inrcinr 7386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-er 6807  df-en 7023  df-dju 7378  df-inl 7387  df-inr 7388

This theorem is used by:  djuenun  7568  exmidunben  13317  enctlem  13323