Theorem pm54.43 8169

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 8137), 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 8168. 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 8345 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 6926	. . . . . . . 8 |- 1o e. On
2	1	elexi 2981	. . . . . . 7 |- 1o e. _V
3	2	ensn1 7372	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 7358	. . . . 5 |- 1o ~~ { 1o }
5		entr 7360	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 671	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4824	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3935	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 209	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 7391	. . . . . 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 6920	. . . . 5 |- 2o = suc 1o
15		df-suc 4724	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2462	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4299	. . 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 7375	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 7375	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		unidm 3498	. . . . . . . . . . . . . 14 |- ( { x } u. { x } ) = { x }
22		sneq 3886	. . . . . . . . . . . . . . 15 |- ( x = y -> { x } = { y } )
23	22	uneq2d 3509	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
24	21, 23	syl5reqr 2489	. . . . . . . . . . . . 13 |- ( x = y -> ( { x } u. { y } ) = { x } )
25		vex 2974	. . . . . . . . . . . . . . 15 |- x e. _V
26	25	ensn1 7372	. . . . . . . . . . . . . 14 |- { x } ~~ 1o
27		1sdom2 7510	. . . . . . . . . . . . . 14 |- 1o ~< 2o
28		ensdomtr 7446	. . . . . . . . . . . . . 14 |- ( ( { x } ~~ 1o /\ 1o ~< 2o ) -> { x } ~< 2o )
29	26, 27, 28	mp2an 672	. . . . . . . . . . . . 13 |- { x } ~< 2o
30	24, 29	syl6eqbr 4328	. . . . . . . . . . . 12 |- ( x = y -> ( { x } u. { y } ) ~< 2o )
31		sdomnen 7337	. . . . . . . . . . . 12 |- ( ( { x } u. { y } ) ~< 2o -> -. ( { x } u. { y } ) ~~ 2o )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
33	32	necon2ai 2655	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
34		disjsn2 3936	. . . . . . . . . 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 3504	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
38	37	breq1d 4301	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
39		ineq12 3546	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
40	39	eqeq1d 2450	. . . . . . . 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 1690	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimiv 1688	. . . 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 1369   E.wex 1586   e. wcel 1756   =/= wne 2605   u. cun 3325   i^i cin 3326   (/)c0 3636   {csn 3876   class class class wbr 4291   Oncon0 4718   suc csuc 4720   1oc1o 6912   2oc2o 6913   ~~ cen 7306   ~< csdm 7308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-om 6476  df-1o 6919  df-2o 6920  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312

This theorem is referenced by:  pr2nelem  8170  pm110.643  8345  isprm2lem  13769