Theorem endjudisj 7222

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 7079	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2		eninl 7109	. . . 4 |- ( A e. V -> (inl " A ) ~~ A )
3	2	3ad2ant1 1019	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inl " A ) ~~ A )
4		eninr 7110	. . . 4 |- ( B e. W -> (inr " B ) ~~ B )
5	4	3ad2ant2 1020	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inr " B ) ~~ B )
6		djuin 7076	. . . 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 1000	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
9		unen 6829	. . 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 1249	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( A u. B ) )
11	1, 10	eqbrtrrid 4051	1 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158   u. cun 3139   i^i cin 3140   (/)c0 3434   class class class wbr 4015   "cima 4641   ~~ cen 6751   ⊔ cdju 7049  inlcinl 7057  inrcinr 7058

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-er 6548  df-en 6754  df-dju 7050  df-inl 7059  df-inr 7060

This theorem is referenced by:  djuenun  7224  dju0en  7226  exmidunben  12440