Theorem unen 7060

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 6985	. . 3 |- ( A ~~ B <-> E. x x : A -1-1-onto-> B )
2		bren 6985	. . 3 |- ( C ~~ D <-> E. y y : C -1-1-onto-> D )
3		eeanv 1988	. . . 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 2818	. . . . . . . 8 |- x e. _V
5		vex 2818	. . . . . . . 8 |- y e. _V
6	4, 5	unex 4564	. . . . . . 7 |- ( x u. y ) e. _V
7		f1oun 5636	. . . . . . 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 6995	. . . . . . 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 1948	. . . 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 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   u. cun 3211   i^i cin 3212   (/)c0 3510   class class class wbr 4111   -1-1-onto->wf1o 5353   ~~ cen 6975

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-en 6978

This theorem is referenced by:  enpr2d  7066  phplem2  7109  fiunsnnn  7140  unsnfi  7181  endjusym  7389  pm54.43  7489  endjudisj  7519  djuen  7520  frecfzennn  10792  unennn  13165