Theorem pr2cv1 7531

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 6692	. . . 4 |- 2o = { (/) , 1o }
2		ensym 7058	. . . 4 |- ( { A , B } ~~ 2o -> 2o ~~ { A , B } )
3	1, 2	eqbrtrrid 4161	. . 3 |- ( { A , B } ~~ 2o -> { (/) , 1o } ~~ { A , B } )
4		bren 7020	. . 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 2824	. . . . . . 7 |- f e. _V
7		0ex 4255	. . . . . . 7 |- (/) e. _V
8	6, 7	fvex 5710	. . . . . 6 |- ( f ` (/) ) e. _V
9		eleq1 2301	. . . . . 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 6685	. . . . . . . 8 |- 1o e. _V
13	6, 12	fvex 5710	. . . . . . 7 |- ( f ` 1o ) e. _V
14		eleq1 2301	. . . . . . 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 533	. . . . . . . 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 2274	. . . . . . 7 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> ( f ` (/) ) = ( f ` 1o ) )
20		f1of1 5633	. . . . . . . . 9 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> f : { (/) , 1o } -1-1-> { A , B } )
21	20	ad2antrr 492	. . . . . . . 8 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> f : { (/) , 1o } -1-1-> { A , B } )
22	7	prid1 3813	. . . . . . . . 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 3814	. . . . . . . . 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 5965	. . . . . . . 8 |- ( ( f : { (/) , 1o } -1-1-> { A , B } /\ ( (/) e. { (/) , 1o } /\ 1o e. { (/) , 1o } ) ) -> ( ( f ` (/) ) = ( f ` 1o ) -> (/) = 1o ) )
27	21, 23, 25, 26	syl12anc 1276	. . . . . . 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 6695	. . . . . . . 8 |- 1o =/= (/)
30	29	nesymi 2466	. . . . . . 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 629	. . . . 5 |- ( ( ( f : { (/) , 1o } -1-1-onto-> { A , B } /\ ( f ` (/) ) = B ) /\ ( f ` 1o ) = B ) -> A e. _V )
33		f1of 5634	. . . . . . . 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 5835	. . . . . . 7 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` 1o ) e. { A , B } )
36		elpri 3728	. . . . . . 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 810	. . . 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 5835	. . . . 5 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> ( f ` (/) ) e. { A , B } )
42		elpri 3728	. . . . 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 810	. . 3 |- ( f : { (/) , 1o } -1-1-onto-> { A , B } -> A e. _V )
45	44	exlimiv 1651	. 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 720   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {cpr 3706   class class class wbr 4125   -1-1->wf1 5369   -1-1-onto->wf1o 5371   ` cfv 5372   1oc1o 6670   2oc2o 6671   ~~ cen 7010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013

This theorem is referenced by:  pr2cv2  7532  pr2cv  7533