Theorem unen 6782

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 6713	. . 3 |- ( A ~~ B <-> E. x x : A -1-1-onto-> B )
2		bren 6713	. . 3 |- ( C ~~ D <-> E. y y : C -1-1-onto-> D )
3		eeanv 1920	. . . 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 2729	. . . . . . . 8 |- x e. _V
5		vex 2729	. . . . . . . 8 |- y e. _V
6	4, 5	unex 4419	. . . . . . 7 |- ( x u. y ) e. _V
7		f1oun 5452	. . . . . . 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 6720	. . . . . . 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 1884	. . . 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 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   u. cun 3114   i^i cin 3115   (/)c0 3409   class class class wbr 3982   -1-1-onto->wf1o 5187   ~~ cen 6704

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707

This theorem is referenced by:  enpr2d  6783  phplem2  6819  fiunsnnn  6847  unsnfi  6884  endjusym  7061  pm54.43  7146  endjudisj  7166  djuen  7167  frecfzennn  10361  unennn  12330