Theorem pm54.43 6719

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 6118	. . . . . . . 8 |- 1o e. On
2	1	elexi 2622	. . . . . . 7 |- 1o e. _V
3	2	ensn1 6441	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 6427	. . . . 5 |- 1o ~~ { 1o }
5		entr 6429	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 416	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4321	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3478	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 144	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 6461	. . . . . 6 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( ( A i^i B ) = (/) /\ ( 1o i^i { 1o } ) = (/) ) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
11	9, 10	mpanr2 429	. . . . 5 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
12	11	ex 113	. . . 4 |- ( ( A ~~ 1o /\ B ~~ { 1o } ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
13	6, 12	sylan2 280	. . 3 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
14		df-2o 6112	. . . . 5 |- 2o = suc 1o
15		df-suc 4161	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2103	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 3819	. . 3 |- ( ( A u. B ) ~~ 2o <-> ( A u. B ) ~~ ( 1o u. { 1o } ) )
18	13, 17	syl6ibr 160	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ 2o ) )
19		en1 6444	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 6444	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		1nen2 6505	. . . . . . . . . . . . 13 |- -. 1o ~~ 2o
22	21	a1i 9	. . . . . . . . . . . 12 |- ( x = y -> -. 1o ~~ 2o )
23		unidm 3127	. . . . . . . . . . . . . . . 16 |- ( { x } u. { x } ) = { x }
24		sneq 3433	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
25	24	uneq2d 3138	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
26	23, 25	syl5reqr 2130	. . . . . . . . . . . . . . 15 |- ( x = y -> ( { x } u. { y } ) = { x } )
27		vex 2615	. . . . . . . . . . . . . . . 16 |- x e. _V
28	27	ensn1 6441	. . . . . . . . . . . . . . 15 |- { x } ~~ 1o
29	26, 28	syl6eqbr 3848	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { y } ) ~~ 1o )
30	29	ensymd 6428	. . . . . . . . . . . . 13 |- ( x = y -> 1o ~~ ( { x } u. { y } ) )
31		entr 6429	. . . . . . . . . . . . 13 |- ( ( 1o ~~ ( { x } u. { y } ) /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
32	30, 31	sylan 277	. . . . . . . . . . . 12 |- ( ( x = y /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
33	22, 32	mtand 624	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
34	33	necon2ai 2303	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
35		disjsn2 3479	. . . . . . . . . 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 3133	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
39	38	breq1d 3821	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
40		ineq12 3180	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
41	40	eqeq1d 2091	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A i^i B ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
42	37, 39, 41	3imtr4d 201	. . . . . . 7 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
43	42	ex 113	. . . . . 6 |- ( A = { x } -> ( B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimdv 1742	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	exlimiv 1530	. . . 4 |- ( E. x A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
46	45	imp 122	. . 3 |- ( ( E. x A = { x } /\ E. y B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
47	19, 20, 46	syl2anb 285	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
48	18, 47	impbid 127	1 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   =/= wne 2249   u. cun 2982   i^i cin 2983   (/)c0 3269   {csn 3422   class class class wbr 3811   Oncon0 4153   suc csuc 4155   1oc1o 6104   2oc2o 6105   ~~ cen 6383

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-1o 6111  df-2o 6112  df-er 6220  df-en 6386

This theorem is referenced by:  pr2nelem  6720  dju1p1e2  6724