Theorem pm54.43 7359

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 6567	. . . . . . . 8 |- 1o e. On
2	1	elexi 2812	. . . . . . 7 |- 1o e. _V
3	2	ensn1 6946	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 6932	. . . . 5 |- 1o ~~ { 1o }
5		entr 6934	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 425	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4634	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3728	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 146	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 6967	. . . . . 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 6561	. . . . 5 |- 2o = suc 1o
15		df-suc 4461	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2250	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4090	. . 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 6949	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 6949	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		1nen2 7018	. . . . . . . . . . . . 13 |- -. 1o ~~ 2o
22	21	a1i 9	. . . . . . . . . . . 12 |- ( x = y -> -. 1o ~~ 2o )
23		sneq 3677	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
24	23	uneq2d 3358	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
25		unidm 3347	. . . . . . . . . . . . . . . 16 |- ( { x } u. { x } ) = { x }
26	24, 25	eqtr3di 2277	. . . . . . . . . . . . . . 15 |- ( x = y -> ( { x } u. { y } ) = { x } )
27		vex 2802	. . . . . . . . . . . . . . . 16 |- x e. _V
28	27	ensn1 6946	. . . . . . . . . . . . . . 15 |- { x } ~~ 1o
29	26, 28	eqbrtrdi 4121	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { y } ) ~~ 1o )
30	29	ensymd 6933	. . . . . . . . . . . . 13 |- ( x = y -> 1o ~~ ( { x } u. { y } ) )
31		entr 6934	. . . . . . . . . . . . 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 669	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
34	33	necon2ai 2454	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
35		disjsn2 3729	. . . . . . . . . 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 3353	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
39	38	breq1d 4092	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
40		ineq12 3400	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
41	40	eqeq1d 2238	. . . . . . . 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 1865	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	exlimiv 1644	. . . 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 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   class class class wbr 4082   Oncon0 4453   suc csuc 4455   1oc1o 6553   2oc2o 6554   ~~ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886

This theorem is referenced by:  pr2nelem  7360  dju1p1e2  7371