Theorem pm54.43 7394

Description: Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)

Assertion

Ref	Expression
pm54.43	|- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Proof of Theorem pm54.43

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

Step	Hyp	Ref	Expression
1		1on 6588	. . . . . . . 8 |- 1o e. On
2	1	elexi 2815	. . . . . . 7 |- 1o e. _V
3	2	ensn1 6969	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 6955	. . . . 5 |- 1o ~~ { 1o }
5		entr 6957	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 425	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4641	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3731	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 146	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 6990	. . . . . 6 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( ( A i^i B ) = (/) /\ ( 1o i^i { 1o } ) = (/) ) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
11	9, 10	mpanr2 438	. . . . 5 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
12	11	ex 115	. . . 4 |- ( ( A ~~ 1o /\ B ~~ { 1o } ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
13	6, 12	sylan2 286	. . 3 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
14		df-2o 6582	. . . . 5 |- 2o = suc 1o
15		df-suc 4468	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2252	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4096	. . 3 |- ( ( A u. B ) ~~ 2o <-> ( A u. B ) ~~ ( 1o u. { 1o } ) )
18	13, 17	imbitrrdi 162	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ 2o ) )
19		en1 6972	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 6972	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		1nen2 7046	. . . . . . . . . . . . 13 |- -. 1o ~~ 2o
22	21	a1i 9	. . . . . . . . . . . 12 |- ( x = y -> -. 1o ~~ 2o )
23		sneq 3680	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
24	23	uneq2d 3361	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
25		unidm 3350	. . . . . . . . . . . . . . . 16 |- ( { x } u. { x } ) = { x }
26	24, 25	eqtr3di 2279	. . . . . . . . . . . . . . 15 |- ( x = y -> ( { x } u. { y } ) = { x } )
27		vex 2805	. . . . . . . . . . . . . . . 16 |- x e. _V
28	27	ensn1 6969	. . . . . . . . . . . . . . 15 |- { x } ~~ 1o
29	26, 28	eqbrtrdi 4127	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { y } ) ~~ 1o )
30	29	ensymd 6956	. . . . . . . . . . . . 13 |- ( x = y -> 1o ~~ ( { x } u. { y } ) )
31		entr 6957	. . . . . . . . . . . . 13 |- ( ( 1o ~~ ( { x } u. { y } ) /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
32	30, 31	sylan 283	. . . . . . . . . . . 12 |- ( ( x = y /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
33	22, 32	mtand 671	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
34	33	necon2ai 2456	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
35		disjsn2 3732	. . . . . . . . . 10 |- ( x =/= y -> ( { x } i^i { y } ) = (/) )
36	34, 35	syl 14	. . . . . . . . 9 |- ( ( { x } u. { y } ) ~~ 2o -> ( { x } i^i { y } ) = (/) )
37	36	a1i 9	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( { x } u. { y } ) ~~ 2o -> ( { x } i^i { y } ) = (/) ) )
38		uneq12 3356	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
39	38	breq1d 4098	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
40		ineq12 3403	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
41	40	eqeq1d 2240	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A i^i B ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
42	37, 39, 41	3imtr4d 203	. . . . . . 7 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
43	42	ex 115	. . . . . 6 |- ( A = { x } -> ( B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimdv 1867	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	exlimiv 1646	. . . 4 |- ( E. x A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
46	45	imp 124	. . 3 |- ( ( E. x A = { x } /\ E. y B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
47	19, 20, 46	syl2anb 291	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
48	18, 47	impbid 129	1 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   =/= wne 2402   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   class class class wbr 4088   Oncon0 4460   suc csuc 4462   1oc1o 6574   2oc2o 6575   ~~ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909

This theorem is referenced by:  pr2nelem  7395  dju1p1e2  7407