Theorem unen 6875

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 6806	. . 3 |- ( A ~~ B <-> E. x x : A -1-1-onto-> B )
2		bren 6806	. . 3 |- ( C ~~ D <-> E. y y : C -1-1-onto-> D )
3		eeanv 1951	. . . 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 2766	. . . . . . . 8 |- x e. _V
5		vex 2766	. . . . . . . 8 |- y e. _V
6	4, 5	unex 4476	. . . . . . 7 |- ( x u. y ) e. _V
7		f1oun 5524	. . . . . . 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 6813	. . . . . . 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 1911	. . . 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 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   u. cun 3155   i^i cin 3156   (/)c0 3450   class class class wbr 4033   -1-1-onto->wf1o 5257   ~~ cen 6797

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-en 6800

This theorem is referenced by:  enpr2d  6876  phplem2  6914  fiunsnnn  6942  unsnfi  6980  endjusym  7162  pm54.43  7257  endjudisj  7277  djuen  7278  frecfzennn  10518  unennn  12614