Theorem djuen 7067

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 6640	. . . . . . . 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 6982	. . . . . 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 6678	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ B ) -> (inl " A ) ~~ B )
8	5, 6, 7	syl2anc 408	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ B )
9		eninl 6982	. . . . . 6 |- ( B e. _V -> (inl " B ) ~~ B )
10	2, 9	simpl2im 383	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " B ) ~~ B )
11	10	ensymd 6677	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> B ~~ (inl " B ) )
12		entr 6678	. . . 4 |- ( ( (inl " A ) ~~ B /\ B ~~ (inl " B ) ) -> (inl " A ) ~~ (inl " B ) )
13	8, 11, 12	syl2anc 408	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inl " A ) ~~ (inl " B ) )
14		encv 6640	. . . . . . . 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 6983	. . . . . 6 |- ( C e. _V -> (inr " C ) ~~ C )
18	16, 17	syl 14	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ C )
19		entr 6678	. . . . 5 |- ( ( (inr " C ) ~~ C /\ C ~~ D ) -> (inr " C ) ~~ D )
20	18, 19	sylancom 416	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ D )
21		eninr 6983	. . . . . 6 |- ( D e. _V -> (inr " D ) ~~ D )
22	15, 21	simpl2im 383	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " D ) ~~ D )
23	22	ensymd 6677	. . . 4 |- ( ( A ~~ B /\ C ~~ D ) -> D ~~ (inr " D ) )
24		entr 6678	. . . 4 |- ( ( (inr " C ) ~~ D /\ D ~~ (inr " D ) ) -> (inr " C ) ~~ (inr " D ) )
25	20, 23, 24	syl2anc 408	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> (inr " C ) ~~ (inr " D ) )
26		djuin 6949	. . . 4 |- ( (inl " A ) i^i (inr " C ) ) = (/)
27	26	a1i 9	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( (inl " A ) i^i (inr " C ) ) = (/) )
28		djuin 6949	. . . 4 |- ( (inl " B ) i^i (inr " D ) ) = (/)
29	28	a1i 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 ) ) )
31	13, 25, 27, 29, 30	syl22anc 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 )
34	31, 32, 33	3brtr3g 3961	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )

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  7068  exmidunben  11939  enctlem  11945