Theorem pm54.43lem 8825

Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8794), so that their A e. 1 means, in our notation, A e. { x | ( card ` x ) = 1o }. Here we show that this is equivalent to A ~~ 1o so that we can use the latter more convenient notation in pm54.43 8826. (Contributed by NM, 4-Nov-2013.)

Assertion

Ref	Expression
pm54.43lem	|- ( A ~~ 1o <-> A e. { x | ( card ` x ) = 1o } )

Distinct variable group:   x, A

Proof of Theorem pm54.43lem

Step	Hyp	Ref	Expression
1		carden2b 8793	. . . 4 |- ( A ~~ 1o -> ( card ` A ) = ( card ` 1o ) )
2		1onn 7719	. . . . 5 |- 1o e. om
3		cardnn 8789	. . . . 5 |- ( 1o e. om -> ( card ` 1o ) = 1o )
4	2, 3	ax-mp 5	. . . 4 |- ( card ` 1o ) = 1o
5	1, 4	syl6eq 2672	. . 3 |- ( A ~~ 1o -> ( card ` A ) = 1o )
6	4	eqeq2i 2634	. . . . 5 |- ( ( card ` A ) = ( card ` 1o ) <-> ( card ` A ) = 1o )
7	6	biimpri 218	. . . 4 |- ( ( card ` A ) = 1o -> ( card ` A ) = ( card ` 1o ) )
8		1n0 7575	. . . . . . . 8 |- 1o =/= (/)
9	8	neii 2796	. . . . . . 7 |- -. 1o = (/)
10		eqeq1 2626	. . . . . . 7 |- ( ( card ` A ) = 1o -> ( ( card ` A ) = (/) <-> 1o = (/) ) )
11	9, 10	mtbiri 317	. . . . . 6 |- ( ( card ` A ) = 1o -> -. ( card ` A ) = (/) )
12		ndmfv 6218	. . . . . 6 |- ( -. A e. dom card -> ( card ` A ) = (/) )
13	11, 12	nsyl2 142	. . . . 5 |- ( ( card ` A ) = 1o -> A e. dom card )
14		1on 7567	. . . . . 6 |- 1o e. On
15		onenon 8775	. . . . . 6 |- ( 1o e. On -> 1o e. dom card )
16	14, 15	ax-mp 5	. . . . 5 |- 1o e. dom card
17		carden2 8813	. . . . 5 |- ( ( A e. dom card /\ 1o e. dom card ) -> ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o ) )
18	13, 16, 17	sylancl 694	. . . 4 |- ( ( card ` A ) = 1o -> ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o ) )
19	7, 18	mpbid 222	. . 3 |- ( ( card ` A ) = 1o -> A ~~ 1o )
20	5, 19	impbii 199	. 2 |- ( A ~~ 1o <-> ( card ` A ) = 1o )
21		elex 3212	. . . 4 |- ( A e. dom card -> A e. _V )
22	13, 21	syl 17	. . 3 |- ( ( card ` A ) = 1o -> A e. _V )
23		fveq2 6191	. . . 4 |- ( x = A -> ( card ` x ) = ( card ` A ) )
24	23	eqeq1d 2624	. . 3 |- ( x = A -> ( ( card ` x ) = 1o <-> ( card ` A ) = 1o ) )
25	22, 24	elab3 3358	. 2 |- ( A e. { x | ( card ` x ) = 1o } <-> ( card ` A ) = 1o )
26	20, 25	bitr4i 267	1 |- ( A ~~ 1o <-> A e. { x | ( card ` x ) = 1o } )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   1oc1o 7553   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by: (None)