Theorem endjudisj 7424

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 7265	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2		eninl 7295	. . . 4 |- ( A e. V -> (inl " A ) ~~ A )
3	2	3ad2ant1 1044	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inl " A ) ~~ A )
4		eninr 7296	. . . 4 |- ( B e. W -> (inr " B ) ~~ B )
5	4	3ad2ant2 1045	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inr " B ) ~~ B )
6		djuin 7262	. . . 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 1025	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
9		unen 6990	. . 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 1274	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( A u. B ) )
11	1, 10	eqbrtrrid 4124	1 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   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  dju0en  7428  exmidunben  13046