Theorem pr2cv1 7399

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 6596	. . . 4 |- 2o = { (/) , 1o }
2		ensym 6954	. . . 4 |- ( { A , B } ~~ 2o -> 2o ~~ { A , B } )
3	1, 2	eqbrtrrid 4124	. . 3 |- ( { A , B } ~~ 2o -> { (/) , 1o } ~~ { A , B } )
4		bren 6916	. . 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 2805	. . . . . . 7 |- f e. _V
7		0ex 4216	. . . . . . 7 |- (/) e. _V
8	6, 7	fvex 5659	. . . . . 6 |- ( f ` (/) ) e. _V
9		eleq1 2294	. . . . . 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 6589	. . . . . . . 8 |- 1o e. _V
13	6, 12	fvex 5659	. . . . . . 7 |- ( f ` 1o ) e. _V
14		eleq1 2294	. . . . . . 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 529	. . . . . . . 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 2267	. . . . . . 7 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( f ` (/) ) = ( f ` 1o ) )
20		f1of1 5582	. . . . . . . . 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 3777	. . . . . . . . 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 3778	. . . . . . . . 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 5909	. . . . . . . 8 |- ( ( f : { (/) , 1o } -1-1-> { A , B } /\ ( (/) e. { (/) , 1o } /\ 1o e. { (/) , 1o } ) ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
27	21, 23, 25, 26	syl12anc 1271	. . . . . . 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 6599	. . . . . . . 8 |- 1o =/= (/)
30	29	nesymi 2448	. . . . . . 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 625	. . . . 5 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> A e. _V )
33		f1of 5583	. . . . . . . 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 5783	. . . . . . 7 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` 1o ) e. { A , B } )
36		elpri 3692	. . . . . . 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 805	. . . 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 5783	. . . . 5 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` (/) ) e. { A , B } )
42		elpri 3692	. . . . 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 805	. . 3 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> A e. _V )
45	44	exlimiv 1646	. 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 715   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   (/)c0 3494   {cpr 3670   class class class wbr 4088   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326   1oc1o 6574   2oc2o 6575   ~~ cen 6906

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909

This theorem is referenced by:  pr2cv2  7400  pr2cv  7401