Theorem unen 6718

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 6649	. . 3 |- ( A ~~ B <-> E. x x : A -1-1-onto-> B )
2		bren 6649	. . 3 |- ( C ~~ D <-> E. y y : C -1-1-onto-> D )
3		eeanv 1905	. . . 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 2692	. . . . . . . 8 |- x e. _V
5		vex 2692	. . . . . . . 8 |- y e. _V
6	4, 5	unex 4370	. . . . . . 7 |- ( x u. y ) e. _V
7		f1oun 5395	. . . . . . 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 6656	. . . . . . 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 411	. . . . . 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 114	. . . . 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 1869	. . . 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 134	. . 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 289	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( A u. C ) ~~ ( B u. D ) ) )
14	13	imp 123	1 |- ( ( ( A ~~ B /\ C ~~ D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( A u. C ) ~~ ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   u. cun 3074   i^i cin 3075   (/)c0 3368   class class class wbr 3937   -1-1-onto->wf1o 5130   ~~ cen 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-en 6643

This theorem is referenced by:  enpr2d  6719  phplem2  6755  fiunsnnn  6783  unsnfi  6815  endjusym  6989  pm54.43  7063  endjudisj  7083  djuen  7084  frecfzennn  10230  unennn  11946