Theorem unen 6818

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem unen

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6749	. . 3 |- ( A ~~ B <-> E. x x : A -1-1-onto-> B )
2		bren 6749	. . 3 |- ( C ~~ D <-> E. y y : C -1-1-onto-> D )
3		eeanv 1932	. . . 4 |- ( E. x E. y ( x : A -1-1-onto-> B /\ y : C -1-1-onto-> D ) <-> ( E. x x : A -1-1-onto-> B /\ E. y y : C -1-1-onto-> D ) )
4		vex 2742	. . . . . . . 8 |- x e. _V
5		vex 2742	. . . . . . . 8 |- y e. _V
6	4, 5	unex 4443	. . . . . . 7 |- ( x u. y ) e. _V
7		f1oun 5483	. . . . . . 7 |- ( ( ( x : A -1-1-onto-> B /\ y : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( x u. y ) : ( A u. C ) -1-1-onto-> ( B u. D ) )
8		f1oen3g 6756	. . . . . . 7 |- ( ( ( x u. y ) e. _V /\ ( x u. y ) : ( A u. C ) -1-1-onto-> ( B u. D ) ) -> ( A u. C ) ~~ ( B u. D ) )
9	6, 7, 8	sylancr 414	. . . . . 6 |- ( ( ( x : A -1-1-onto-> B /\ y : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) )
10	9	ex 115	. . . . 5 |- ( ( x : A -1-1-onto-> B /\ y : C -1-1-onto-> D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( A u. C ) ~~ ( B u. D ) ) )
11	10	exlimivv 1896	. . . 4 |- ( E. x E. y ( x : A -1-1-onto-> B /\ y : C -1-1-onto-> D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( A u. C ) ~~ ( B u. D ) ) )
12	3, 11	sylbir 135	. . 3 |- ( ( E. x x : A -1-1-onto-> B /\ E. y y : C -1-1-onto-> D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( A u. C ) ~~ ( B u. D ) ) )
13	1, 2, 12	syl2anb 291	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( A u. C ) ~~ ( B u. D ) ) )
14	13	imp 124	1 |- ( ( ( A ~~ B /\ C ~~ D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   u. cun 3129   i^i cin 3130   (/)c0 3424   class class class wbr 4005   -1-1-onto->wf1o 5217   ~~ cen 6740

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-en 6743

This theorem is referenced by:  enpr2d  6819  phplem2  6855  fiunsnnn  6883  unsnfi  6920  endjusym  7097  pm54.43  7191  endjudisj  7211  djuen  7212  frecfzennn  10428  unennn  12400