Theorem pm54.43 7104

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 6360	. . . . . . . 8 |- 1o e. On
2	1	elexi 2721	. . . . . . 7 |- 1o e. _V
3	2	ensn1 6730	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 6716	. . . . 5 |- 1o ~~ { 1o }
5		entr 6718	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 422	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4496	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3617	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 145	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 6750	. . . . . 6 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( ( A i^i B ) = (/) /\ ( 1o i^i { 1o } ) = (/) ) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
11	9, 10	mpanr2 435	. . . . 5 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
12	11	ex 114	. . . 4 |- ( ( A ~~ 1o /\ B ~~ { 1o } ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
13	6, 12	sylan2 284	. . 3 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
14		df-2o 6354	. . . . 5 |- 2o = suc 1o
15		df-suc 4326	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2175	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 3969	. . 3 |- ( ( A u. B ) ~~ 2o <-> ( A u. B ) ~~ ( 1o u. { 1o } ) )
18	13, 17	syl6ibr 161	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ 2o ) )
19		en1 6733	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 6733	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		1nen2 6795	. . . . . . . . . . . . 13 |- -. 1o ~~ 2o
22	21	a1i 9	. . . . . . . . . . . 12 |- ( x = y -> -. 1o ~~ 2o )
23		unidm 3246	. . . . . . . . . . . . . . . 16 |- ( { x } u. { x } ) = { x }
24		sneq 3567	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
25	24	uneq2d 3257	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
26	23, 25	syl5reqr 2202	. . . . . . . . . . . . . . 15 |- ( x = y -> ( { x } u. { y } ) = { x } )
27		vex 2712	. . . . . . . . . . . . . . . 16 |- x e. _V
28	27	ensn1 6730	. . . . . . . . . . . . . . 15 |- { x } ~~ 1o
29	26, 28	eqbrtrdi 3999	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { y } ) ~~ 1o )
30	29	ensymd 6717	. . . . . . . . . . . . 13 |- ( x = y -> 1o ~~ ( { x } u. { y } ) )
31		entr 6718	. . . . . . . . . . . . 13 |- ( ( 1o ~~ ( { x } u. { y } ) /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
32	30, 31	sylan 281	. . . . . . . . . . . 12 |- ( ( x = y /\ ( { x } u. { y } ) ~~ 2o ) -> 1o ~~ 2o )
33	22, 32	mtand 655	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
34	33	necon2ai 2378	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
35		disjsn2 3618	. . . . . . . . . 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 3252	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
39	38	breq1d 3971	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
40		ineq12 3299	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
41	40	eqeq1d 2163	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A i^i B ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
42	37, 39, 41	3imtr4d 202	. . . . . . 7 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
43	42	ex 114	. . . . . 6 |- ( A = { x } -> ( B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimdv 1796	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	exlimiv 1575	. . . 4 |- ( E. x A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
46	45	imp 123	. . 3 |- ( ( E. x A = { x } /\ E. y B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
47	19, 20, 46	syl2anb 289	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
48	18, 47	impbid 128	1 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 2125   =/= wne 2324   u. cun 3096   i^i cin 3097   (/)c0 3390   {csn 3556   class class class wbr 3961   Oncon0 4318   suc csuc 4320   1oc1o 6346   2oc2o 6347   ~~ cen 6672

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-1o 6353  df-2o 6354  df-er 6469  df-en 6675

This theorem is referenced by:  pr2nelem  7105  dju1p1e2  7111