Theorem pm54.43 8381

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 8349), 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 8380. 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 8557 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 7137	. . . . . . . 8 |- 1o e. On
2	1	elexi 3123	. . . . . . 7 |- 1o e. _V
3	2	ensn1 7579	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 7565	. . . . 5 |- 1o ~~ { 1o }
5		entr 7567	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 671	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4984	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 4088	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 209	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 7598	. . . . . 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 7131	. . . . 5 |- 2o = suc 1o
15		df-suc 4884	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2496	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4455	. . 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 7582	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 7582	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		unidm 3647	. . . . . . . . . . . . . 14 |- ( { x } u. { x } ) = { x }
22		sneq 4037	. . . . . . . . . . . . . . 15 |- ( x = y -> { x } = { y } )
23	22	uneq2d 3658	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
24	21, 23	syl5reqr 2523	. . . . . . . . . . . . 13 |- ( x = y -> ( { x } u. { y } ) = { x } )
25		vex 3116	. . . . . . . . . . . . . . 15 |- x e. _V
26	25	ensn1 7579	. . . . . . . . . . . . . 14 |- { x } ~~ 1o
27		1sdom2 7718	. . . . . . . . . . . . . 14 |- 1o ~< 2o
28		ensdomtr 7653	. . . . . . . . . . . . . 14 |- ( ( { x } ~~ 1o /\ 1o ~< 2o ) -> { x } ~< 2o )
29	26, 27, 28	mp2an 672	. . . . . . . . . . . . 13 |- { x } ~< 2o
30	24, 29	syl6eqbr 4484	. . . . . . . . . . . 12 |- ( x = y -> ( { x } u. { y } ) ~< 2o )
31		sdomnen 7544	. . . . . . . . . . . 12 |- ( ( { x } u. { y } ) ~< 2o -> -. ( { x } u. { y } ) ~~ 2o )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
33	32	necon2ai 2702	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
34		disjsn2 4089	. . . . . . . . . 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 3653	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
38	37	breq1d 4457	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
39		ineq12 3695	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
40	39	eqeq1d 2469	. . . . . . . 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 1700	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimiv 1698	. . . 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 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   u. cun 3474   i^i cin 3475   (/)c0 3785   {csn 4027   class class class wbr 4447   Oncon0 4878   suc csuc 4880   1oc1o 7123   2oc2o 7124   ~~ cen 7513   ~< csdm 7515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-2o 7131  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519

This theorem is referenced by:  pr2nelem  8382  pm110.643  8557  isprm2lem  14083