Theorem pm54.43 8398

Description: Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8366), so that their A e. 1 means, in our notation, A e. { x | ( card ` x ) = 1o } which is the same as A ~~ 1o by pm54.43lem 8397. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 8574 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (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 7155	. . . . . . . 8 |- 1o e. On
2	1	elexi 3119	. . . . . . 7 |- 1o e. _V
3	2	ensn1 7598	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 7584	. . . . 5 |- 1o ~~ { 1o }
5		entr 7586	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 671	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4993	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 4092	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 209	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 7617	. . . . . 6 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( ( A i^i B ) = (/) /\ ( 1o i^i { 1o } ) = (/) ) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
11	9, 10	mpanr2 684	. . . . 5 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
12	11	ex 434	. . . 4 |- ( ( A ~~ 1o /\ B ~~ { 1o } ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
13	6, 12	sylan2 474	. . 3 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
14		df-2o 7149	. . . . 5 |- 2o = suc 1o
15		df-suc 4893	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2486	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4464	. . 3 |- ( ( A u. B ) ~~ 2o <-> ( A u. B ) ~~ ( 1o u. { 1o } ) )
18	13, 17	syl6ibr 227	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ 2o ) )
19		en1 7601	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 7601	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		unidm 3643	. . . . . . . . . . . . . 14 |- ( { x } u. { x } ) = { x }
22		sneq 4042	. . . . . . . . . . . . . . 15 |- ( x = y -> { x } = { y } )
23	22	uneq2d 3654	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
24	21, 23	syl5reqr 2513	. . . . . . . . . . . . 13 |- ( x = y -> ( { x } u. { y } ) = { x } )
25		vex 3112	. . . . . . . . . . . . . . 15 |- x e. _V
26	25	ensn1 7598	. . . . . . . . . . . . . 14 |- { x } ~~ 1o
27		1sdom2 7737	. . . . . . . . . . . . . 14 |- 1o ~< 2o
28		ensdomtr 7672	. . . . . . . . . . . . . 14 |- ( ( { x } ~~ 1o /\ 1o ~< 2o ) -> { x } ~< 2o )
29	26, 27, 28	mp2an 672	. . . . . . . . . . . . 13 |- { x } ~< 2o
30	24, 29	syl6eqbr 4493	. . . . . . . . . . . 12 |- ( x = y -> ( { x } u. { y } ) ~< 2o )
31		sdomnen 7563	. . . . . . . . . . . 12 |- ( ( { x } u. { y } ) ~< 2o -> -. ( { x } u. { y } ) ~~ 2o )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
33	32	necon2ai 2692	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
34		disjsn2 4093	. . . . . . . . . 10 |- ( x =/= y -> ( { x } i^i { y } ) = (/) )
35	33, 34	syl 16	. . . . . . . . 9 |- ( ( { x } u. { y } ) ~~ 2o -> ( { x } i^i { y } ) = (/) )
36	35	a1i 11	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( { x } u. { y } ) ~~ 2o -> ( { x } i^i { y } ) = (/) ) )
37		uneq12 3649	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
38	37	breq1d 4466	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
39		ineq12 3691	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
40	39	eqeq1d 2459	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A i^i B ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
41	36, 38, 40	3imtr4d 268	. . . . . . 7 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
42	41	ex 434	. . . . . 6 |- ( A = { x } -> ( B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
43	42	exlimdv 1725	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimiv 1723	. . . 4 |- ( E. x A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	imp 429	. . 3 |- ( ( E. x A = { x } /\ E. y B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
46	19, 20, 45	syl2anb 479	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
47	18, 46	impbid 191	1 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   u. cun 3469   i^i cin 3470   (/)c0 3793   {csn 4032   class class class wbr 4456   Oncon0 4887   suc csuc 4889   1oc1o 7141   2oc2o 7142   ~~ cen 7532   ~< csdm 7534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538

This theorem is referenced by:  pr2nelem  8399  pm110.643  8574  isprm2lem  14236