Theorem djuenun 7487

Description: Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)

Assertion

Ref	Expression
djuenun	|- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( A ⊔ C ) ~~ ( B u. D ) )

Proof of Theorem djuenun

Step	Hyp	Ref	Expression
1		djuen 7486	. . 3 |- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
2	1	3adant3 1044	. 2 |- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
3		relen 6956	. . . 4 |- Rel ~~
4	3	brrelex2i 4776	. . 3 |- ( A ~~ B -> B e. _V )
5	3	brrelex2i 4776	. . 3 |- ( C ~~ D -> D e. _V )
6		id 19	. . 3 |- ( ( B i^i D ) = (/) -> ( B i^i D ) = (/) )
7		endjudisj 7485	. . 3 |- ( ( B e. _V /\ D e. _V /\ ( B i^i D ) = (/) ) -> ( B ⊔ D ) ~~ ( B u. D ) )
8	4, 5, 6, 7	syl3an 1316	. 2 |- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( B ⊔ D ) ~~ ( B u. D ) )
9		entr 7001	. 2 |- ( ( ( A ⊔ C ) ~~ ( B ⊔ D ) /\ ( B ⊔ D ) ~~ ( B u. D ) ) -> ( A ⊔ C ) ~~ ( B u. D ) )
10	2, 8, 9	syl2anc 411	1 |- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( A ⊔ C ) ~~ ( B u. D ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   u. cun 3199   i^i cin 3200   (/)c0 3496   class class class wbr 4093   ~~ cen 6950   ⊔ cdju 7296

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-er 6745  df-en 6953  df-dju 7297  df-inl 7306  df-inr 7307

This theorem is referenced by:  dju1en  7488  djucomen  7491  djuassen  7492  xpdjuen  7493