Theorem endjudisj 7392

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 7234	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2		eninl 7264	. . . 4 |- ( A e. V -> (inl " A ) ~~ A )
3	2	3ad2ant1 1042	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inl " A ) ~~ A )
4		eninr 7265	. . . 4 |- ( B e. W -> (inr " B ) ~~ B )
5	4	3ad2ant2 1043	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> (inr " B ) ~~ B )
6		djuin 7231	. . . 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 1023	. . 3 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
9		unen 6969	. . 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 1272	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( A u. B ) )
11	1, 10	eqbrtrrid 4119	1 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   u. cun 3195   i^i cin 3196   (/)c0 3491   class class class wbr 4083   "cima 4722   ~~ cen 6885   ⊔ cdju 7204  inlcinl 7212  inrcinr 7213

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-er 6680  df-en 6888  df-dju 7205  df-inl 7214  df-inr 7215

This theorem is referenced by:  djuenun  7394  dju0en  7396  exmidunben  12997