Theorem endjudisj 7277

Description: Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)

Assertion

Ref	Expression
endjudisj	|- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )

Proof of Theorem endjudisj

Step	Hyp	Ref	Expression
1		djuun 7133	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2		eninl 7163	. . . 4 |- ( A e. V -> (inl " A ) ~~ A )
3	2	3ad2ant1 1020	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inl " A ) ~~ A )
4		eninr 7164	. . . 4 |- ( B e. W -> (inr " B ) ~~ B )
5	4	3ad2ant2 1021	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inr " B ) ~~ B )
6		djuin 7130	. . . 4 |- ( (inl " A ) i^i (inr " B ) ) = (/)
7	6	a1i 9	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( (inl " A ) i^i (inr " B ) ) = (/) )
8		simp3 1001	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
9		unen 6875	. . 3 |- ( ( ( (inl " A ) ~~ A /\ (inr " B ) ~~ B ) /\ ( ( (inl " A ) i^i (inr " B ) ) = (/) /\ ( A i^i B ) = (/) ) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( A u. B ) )
10	3, 5, 7, 8, 9	syl22anc 1250	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( A u. B ) )
11	1, 10	eqbrtrrid 4069	1 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   u. cun 3155   i^i cin 3156   (/)c0 3450   class class class wbr 4033   "cima 4666   ~~ cen 6797   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-er 6592  df-en 6800  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  djuenun  7279  dju0en  7281  exmidunben  12643