Theorem pr2cv1 7324

Description: If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)

Assertion

Ref	Expression
pr2cv1	|- ( { A , B } ~~ 2o -> A e. _V )

Proof of Theorem pr2cv1

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2o3 6534	. . . 4 |- 2o = { (/) , 1o }
2		ensym 6891	. . . 4 |- ( { A , B } ~~ 2o -> 2o ~~ { A , B } )
3	1, 2	eqbrtrrid 4090	. . 3 |- ( { A , B } ~~ 2o -> { (/) , 1o } ~~ { A , B } )
4		bren 6853	. . 3 |- ( { (/) , 1o } ~~ { A , B } <-> E. f f : { (/) , 1o } -1-1-onto-> { A , B } )
5	3, 4	sylib 122	. 2 |- ( { A , B } ~~ 2o -> E. f f : { (/) , 1o } -1-1-onto-> { A , B } )
6		vex 2776	. . . . . . 7 |- f e. _V
7		0ex 4182	. . . . . . 7 |- (/) e. _V
8	6, 7	fvex 5614	. . . . . 6 |- ( f ` (/) ) e. _V
9		eleq1 2269	. . . . . 6 |- ( ( f ` (/) ) = A -> ( ( f ` (/) ) e. _V <-> A e. _V ) )
10	8, 9	mpbii 148	. . . . 5 |- ( ( f ` (/) ) = A -> A e. _V )
11	10	adantl 277	. . . 4 |- ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = A ) -> A e. _V )
12		1oex 6528	. . . . . . . 8 |- 1o e. _V
13	6, 12	fvex 5614	. . . . . . 7 |- ( f ` 1o ) e. _V
14		eleq1 2269	. . . . . . 7 |- ( ( f ` 1o ) = A -> ( ( f ` 1o ) e. _V <-> A e. _V ) )
15	13, 14	mpbii 148	. . . . . 6 |- ( ( f ` 1o ) = A -> A e. _V )
16	15	adantl 277	. . . . 5 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = A ) -> A e. _V )
17		simplr 528	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( f ` (/) ) = B )
18		simpr 110	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( f ` 1o ) = B )
19	17, 18	eqtr4d 2242	. . . . . . 7 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( f ` (/) ) = ( f ` 1o ) )
20		f1of1 5538	. . . . . . . . 9 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> f : { (/) , 1o } -1-1-> { A , B } )
21	20	ad2antrr 488	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> f : { (/) , 1o } -1-1-> { A , B } )
22	7	prid1 3744	. . . . . . . . 9 |- (/) e. { (/) , 1o }
23	22	a1i 9	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> (/) e. { (/) , 1o } )
24	12	prid2 3745	. . . . . . . . 9 |- 1o e. { (/) , 1o }
25	24	a1i 9	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> 1o e. { (/) , 1o } )
26		f1veqaeq 5856	. . . . . . . 8 |- ( ( f : { (/) , 1o } -1-1-> { A , B } /\ ( (/) e. { (/) , 1o } /\ 1o e. { (/) , 1o } ) ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
27	21, 23, 25, 26	syl12anc 1248	. . . . . . 7 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
28	19, 27	mpd 13	. . . . . 6 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> (/) = 1o )
29		1n0 6536	. . . . . . . 8 |- 1o =/= (/)
30	29	nesymi 2423	. . . . . . 7 |- -. (/) = 1o
31	30	a1i 9	. . . . . 6 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> -. (/) = 1o )
32	28, 31	pm2.21dd 621	. . . . 5 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> A e. _V )
33		f1of 5539	. . . . . . . 8 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> f : { (/) , 1o } --> { A , B } )
34	24	a1i 9	. . . . . . . 8 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> 1o e. { (/) , 1o } )
35	33, 34	ffvelcdmd 5734	. . . . . . 7 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` 1o ) e. { A , B } )
36		elpri 3661	. . . . . . 7 |- ( ( f ` 1o ) e. { A , B } -> ( ( f ` 1o ) = A \/ ( f ` 1o ) = B ) )
37	35, 36	syl 14	. . . . . 6 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( ( f ` 1o ) = A \/ ( f ` 1o ) = B ) )
38	37	adantr 276	. . . . 5 |- ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) -> ( ( f ` 1o ) = A \/ ( f ` 1o ) = B ) )
39	16, 32, 38	mpjaodan 800	. . . 4 |- ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) -> A e. _V )
40	22	a1i 9	. . . . . 6 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> (/) e. { (/) , 1o } )
41	33, 40	ffvelcdmd 5734	. . . . 5 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` (/) ) e. { A , B } )
42		elpri 3661	. . . . 5 |- ( ( f ` (/) ) e. { A , B } -> ( ( f ` (/) ) = A \/ ( f ` (/) ) = B ) )
43	41, 42	syl 14	. . . 4 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( ( f ` (/) ) = A \/ ( f ` (/) ) = B ) )
44	11, 39, 43	mpjaodan 800	. . 3 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> A e. _V )
45	44	exlimiv 1622	. 2 |- ( E. f f : { (/) , 1o } -1-1-onto-> { A , B } -> A e. _V )
46	5, 45	syl 14	1 |- ( { A , B } ~~ 2o -> A e. _V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   (/)c0 3464   {cpr 3639   class class class wbr 4054   -1-1->wf1 5282   -1-1-onto->wf1o 5284   ` cfv 5285   1oc1o 6513   2oc2o 6514   ~~ cen 6843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1o 6520  df-2o 6521  df-er 6638  df-en 6846

This theorem is referenced by:  pr2cv2  7325  pr2cv  7326