Theorem djuen 7167

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 6712	. . . . . . . 8 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
2	1	adantr 274	. . . . . . 7 |- ( ( A ~~ B /\ C ~~ D ) -> ( A e. _V /\ B e. _V ) )
3	2	simpld 111	. . . . . 6 |- ( ( A ~~ B /\ C ~~ D ) -> A e. _V )
4		eninl 7062	. . . . . 6 |- ( A e. _V -> (inl " A ) ~~ A )
5	3, 4	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ A )
6		simpl 108	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> A ~~ B )
7		entr 6750	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ B ) -> (inl " A ) ~~ B )
8	5, 6, 7	syl2anc 409	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ B )
9		eninl 7062	. . . . . 6 |- ( B e. _V -> (inl " B ) ~~ B )
10	2, 9	simpl2im 384	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " B ) ~~ B )
11	10	ensymd 6749	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> B ~~ (inl " B ) )
12		entr 6750	. . . 4 |- ( ( (inl " A ) ~~ B /\ B ~~ (inl " B ) ) -> (inl " A ) ~~ (inl " B ) )
13	8, 11, 12	syl2anc 409	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ (inl " B ) )
14		encv 6712	. . . . . . . 8 |- ( C ~~ D -> ( C e. _V /\ D e. _V ) )
15	14	adantl 275	. . . . . . 7 |- ( ( A ~~ B /\ C ~~ D ) -> ( C e. _V /\ D e. _V ) )
16	15	simpld 111	. . . . . 6 |- ( ( A ~~ B /\ C ~~ D ) -> C e. _V )
17		eninr 7063	. . . . . 6 |- ( C e. _V -> (inr " C ) ~~ C )
18	16, 17	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ C )
19		entr 6750	. . . . 5 |- ( ( (inr " C ) ~~ C /\ C ~~ D ) -> (inr " C ) ~~ D )
20	18, 19	sylancom 417	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ D )
21		eninr 7063	. . . . . 6 |- ( D e. _V -> (inr " D ) ~~ D )
22	15, 21	simpl2im 384	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " D ) ~~ D )
23	22	ensymd 6749	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> D ~~ (inr " D ) )
24		entr 6750	. . . 4 |- ( ( (inr " C ) ~~ D /\ D ~~ (inr " D ) ) -> (inr " C ) ~~ (inr " D ) )
25	20, 23, 24	syl2anc 409	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ (inr " D ) )
26		djuin 7029	. . . 4 |- ( (inl " A ) i^i (inr " C ) ) = (/)
27	26	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) i^i (inr " C ) ) = (/) )
28		djuin 7029	. . . 4 |- ( (inl " B ) i^i (inr " D ) ) = (/)
29	28	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " B ) i^i (inr " D ) ) = (/) )
30		unen 6782	. . 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 1229	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) u. (inr " C ) ) ~~ ( (inl " B ) u. (inr " D ) ) )
32		djuun 7032	. 2 |- ( (inl " A ) u. (inr " C ) ) = ( A ⊔ C )
33		djuun 7032	. 2 |- ( (inl " B ) u. (inr " D ) ) = ( B ⊔ D )
34	31, 32, 33	3brtr3g 4015	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   i^i cin 3115   (/)c0 3409   class class class wbr 3982   "cima 4607   ~~ cen 6704   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-er 6501  df-en 6707  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  djuenun  7168  exmidunben  12359  enctlem  12365