Theorem pm54.43 7843

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 7811), 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 7842. 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 8013 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 6690	. . . . . . . 8 |- 1o e. On
2	1	elexi 2925	. . . . . . 7 |- 1o e. _V
3	2	ensn1 7130	. . . . . 6 |- { 1o } ~~ 1o
4	3	ensymi 7116	. . . . 5 |- 1o ~~ { 1o }
5		entr 7118	. . . . 5 |- ( ( B ~~ 1o /\ 1o ~~ { 1o } ) -> B ~~ { 1o } )
6	4, 5	mpan2 653	. . . 4 |- ( B ~~ 1o -> B ~~ { 1o } )
7	1	onirri 4647	. . . . . . 7 |- -. 1o e. 1o
8		disjsn 3828	. . . . . . 7 |- ( ( 1o i^i { 1o } ) = (/) <-> -. 1o e. 1o )
9	7, 8	mpbir 201	. . . . . 6 |- ( 1o i^i { 1o } ) = (/)
10		unen 7148	. . . . . 6 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( ( A i^i B ) = (/) /\ ( 1o i^i { 1o } ) = (/) ) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
11	9, 10	mpanr2 666	. . . . 5 |- ( ( ( A ~~ 1o /\ B ~~ { 1o } ) /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ ( 1o u. { 1o } ) )
12	11	ex 424	. . . 4 |- ( ( A ~~ 1o /\ B ~~ { 1o } ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
13	6, 12	sylan2 461	. . 3 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ ( 1o u. { 1o } ) ) )
14		df-2o 6684	. . . . 5 |- 2o = suc 1o
15		df-suc 4547	. . . . 5 |- suc 1o = ( 1o u. { 1o } )
16	14, 15	eqtri 2424	. . . 4 |- 2o = ( 1o u. { 1o } )
17	16	breq2i 4180	. . 3 |- ( ( A u. B ) ~~ 2o <-> ( A u. B ) ~~ ( 1o u. { 1o } ) )
18	13, 17	syl6ibr 219	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) -> ( A u. B ) ~~ 2o ) )
19		en1 7133	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
20		en1 7133	. . 3 |- ( B ~~ 1o <-> E. y B = { y } )
21		unidm 3450	. . . . . . . . . . . . . 14 |- ( { x } u. { x } ) = { x }
22		sneq 3785	. . . . . . . . . . . . . . 15 |- ( x = y -> { x } = { y } )
23	22	uneq2d 3461	. . . . . . . . . . . . . 14 |- ( x = y -> ( { x } u. { x } ) = ( { x } u. { y } ) )
24	21, 23	syl5reqr 2451	. . . . . . . . . . . . 13 |- ( x = y -> ( { x } u. { y } ) = { x } )
25		vex 2919	. . . . . . . . . . . . . . 15 |- x e. _V
26	25	ensn1 7130	. . . . . . . . . . . . . 14 |- { x } ~~ 1o
27		1sdom2 7266	. . . . . . . . . . . . . 14 |- 1o ~< 2o
28		ensdomtr 7202	. . . . . . . . . . . . . 14 |- ( ( { x } ~~ 1o /\ 1o ~< 2o ) -> { x } ~< 2o )
29	26, 27, 28	mp2an 654	. . . . . . . . . . . . 13 |- { x } ~< 2o
30	24, 29	syl6eqbr 4209	. . . . . . . . . . . 12 |- ( x = y -> ( { x } u. { y } ) ~< 2o )
31		sdomnen 7095	. . . . . . . . . . . 12 |- ( ( { x } u. { y } ) ~< 2o -> -. ( { x } u. { y } ) ~~ 2o )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( x = y -> -. ( { x } u. { y } ) ~~ 2o )
33	32	necon2ai 2612	. . . . . . . . . 10 |- ( ( { x } u. { y } ) ~~ 2o -> x =/= y )
34		disjsn2 3829	. . . . . . . . . 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 3456	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A u. B ) = ( { x } u. { y } ) )
38	37	breq1d 4182	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o <-> ( { x } u. { y } ) ~~ 2o ) )
39		ineq12 3497	. . . . . . . . 9 |- ( ( A = { x } /\ B = { y } ) -> ( A i^i B ) = ( { x } i^i { y } ) )
40	39	eqeq1d 2412	. . . . . . . 8 |- ( ( A = { x } /\ B = { y } ) -> ( ( A i^i B ) = (/) <-> ( { x } i^i { y } ) = (/) ) )
41	36, 38, 40	3imtr4d 260	. . . . . . 7 |- ( ( A = { x } /\ B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
42	41	ex 424	. . . . . 6 |- ( A = { x } -> ( B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
43	42	exlimdv 1643	. . . . 5 |- ( A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
44	43	exlimiv 1641	. . . 4 |- ( E. x A = { x } -> ( E. y B = { y } -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) ) )
45	44	imp 419	. . 3 |- ( ( E. x A = { x } /\ E. y B = { y } ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
46	19, 20, 45	syl2anb 466	. 2 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A u. B ) ~~ 2o -> ( A i^i B ) = (/) ) )
47	18, 46	impbid 184	1 |- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   u. cun 3278   i^i cin 3279   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541   suc csuc 4543   1oc1o 6676   2oc2o 6677   ~~ cen 7065   ~< csdm 7067

This theorem is referenced by:  pr2nelem  7844  pm110.643  8013  isprm2lem  13041

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071