Theorem pr2pwpr 13261

Description: The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)

Assertion

Ref	Expression
pr2pwpr	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { p e. ~P { A , B } | p ~~ 2o } = { { A , B } } )

Distinct variable groups:   A, p   B, p

Allowed substitution hints:   V( p)   W( p)

Proof of Theorem pr2pwpr

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4168	. . . . . . 7 |- ( s e. ~P { A , B } -> s C_ { A , B } )
2		prfi 8235	. . . . . . . . 9 |- { A , B } e. Fin
3		ssfi 8180	. . . . . . . . 9 |- ( ( { A , B } e. Fin /\ s C_ { A , B } ) -> s e. Fin )
4	2, 3	mpan 706	. . . . . . . 8 |- ( s C_ { A , B } -> s e. Fin )
5		hash2 13193	. . . . . . . . . . . . . 14 |- ( # ` 2o ) = 2
6	5	eqcomi 2631	. . . . . . . . . . . . 13 |- 2 = ( # ` 2o )
7	6	a1i 11	. . . . . . . . . . . 12 |- ( s e. Fin -> 2 = ( # ` 2o ) )
8	7	eqeq2d 2632	. . . . . . . . . . 11 |- ( s e. Fin -> ( ( # ` s ) = 2 <-> ( # ` s ) = ( # ` 2o ) ) )
9		2onn 7720	. . . . . . . . . . . . 13 |- 2o e. om
10		nnfi 8153	. . . . . . . . . . . . 13 |- ( 2o e. om -> 2o e. Fin )
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 |- 2o e. Fin
12		hashen 13135	. . . . . . . . . . . 12 |- ( ( s e. Fin /\ 2o e. Fin ) -> ( ( # ` s ) = ( # ` 2o ) <-> s ~~ 2o ) )
13	11, 12	mpan2 707	. . . . . . . . . . 11 |- ( s e. Fin -> ( ( # ` s ) = ( # ` 2o ) <-> s ~~ 2o ) )
14	8, 13	bitrd 268	. . . . . . . . . 10 |- ( s e. Fin -> ( ( # ` s ) = 2 <-> s ~~ 2o ) )
15		hash2pwpr 13258	. . . . . . . . . . . 12 |- ( ( ( # ` s ) = 2 /\ s e. ~P { A , B } ) -> s = { A , B } )
16	15	a1d 25	. . . . . . . . . . 11 |- ( ( ( # ` s ) = 2 /\ s e. ~P { A , B } ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) )
17	16	ex 450	. . . . . . . . . 10 |- ( ( # ` s ) = 2 -> ( s e. ~P { A , B } -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) ) )
18	14, 17	syl6bir 244	. . . . . . . . 9 |- ( s e. Fin -> ( s ~~ 2o -> ( s e. ~P { A , B } -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) ) ) )
19	18	com23 86	. . . . . . . 8 |- ( s e. Fin -> ( s e. ~P { A , B } -> ( s ~~ 2o -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) ) ) )
20	4, 19	syl 17	. . . . . . 7 |- ( s C_ { A , B } -> ( s e. ~P { A , B } -> ( s ~~ 2o -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) ) ) )
21	1, 20	mpcom 38	. . . . . 6 |- ( s e. ~P { A , B } -> ( s ~~ 2o -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) ) )
22	21	imp 445	. . . . 5 |- ( ( s e. ~P { A , B } /\ s ~~ 2o ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> s = { A , B } ) )
23	22	com12 32	. . . 4 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( s e. ~P { A , B } /\ s ~~ 2o ) -> s = { A , B } ) )
24		prex 4909	. . . . . . . . . . . . 13 |- { A , B } e. _V
25	24	prid2 4298	. . . . . . . . . . . 12 |- { A , B } e. { { B } , { A , B } }
26	25	a1i 11	. . . . . . . . . . 11 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. { { B } , { A , B } } )
27	26	olcd 408	. . . . . . . . . 10 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } e. { (/) , { A } } \/ { A , B } e. { { B } , { A , B } } ) )
28		elun 3753	. . . . . . . . . 10 |- ( { A , B } e. ( { (/) , { A } } u. { { B } , { A , B } } ) <-> ( { A , B } e. { (/) , { A } } \/ { A , B } e. { { B } , { A , B } } ) )
29	27, 28	sylibr 224	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. ( { (/) , { A } } u. { { B } , { A , B } } ) )
30		pwpr 4430	. . . . . . . . 9 |- ~P { A , B } = ( { (/) , { A } } u. { { B } , { A , B } } )
31	29, 30	syl6eleqr 2712	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. ~P { A , B } )
32	31	adantr 481	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> { A , B } e. ~P { A , B } )
33		eleq1 2689	. . . . . . . 8 |- ( s = { A , B } -> ( s e. ~P { A , B } <-> { A , B } e. ~P { A , B } ) )
34	33	adantl 482	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> ( s e. ~P { A , B } <-> { A , B } e. ~P { A , B } ) )
35	32, 34	mpbird 247	. . . . . 6 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> s e. ~P { A , B } )
36		pr2nelem 8827	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } ~~ 2o )
37	36	adantr 481	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> { A , B } ~~ 2o )
38		breq1 4656	. . . . . . . 8 |- ( s = { A , B } -> ( s ~~ 2o <-> { A , B } ~~ 2o ) )
39	38	adantl 482	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> ( s ~~ 2o <-> { A , B } ~~ 2o ) )
40	37, 39	mpbird 247	. . . . . 6 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> s ~~ 2o )
41	35, 40	jca 554	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ s = { A , B } ) -> ( s e. ~P { A , B } /\ s ~~ 2o ) )
42	41	ex 450	. . . 4 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( s = { A , B } -> ( s e. ~P { A , B } /\ s ~~ 2o ) ) )
43	23, 42	impbid 202	. . 3 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( s e. ~P { A , B } /\ s ~~ 2o ) <-> s = { A , B } ) )
44		breq1 4656	. . . 4 |- ( p = s -> ( p ~~ 2o <-> s ~~ 2o ) )
45	44	elrab 3363	. . 3 |- ( s e. { p e. ~P { A , B } | p ~~ 2o } <-> ( s e. ~P { A , B } /\ s ~~ 2o ) )
46		velsn 4193	. . 3 |- ( s e. { { A , B } } <-> s = { A , B } )
47	43, 45, 46	3bitr4g 303	. 2 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( s e. { p e. ~P { A , B } | p ~~ 2o } <-> s e. { { A , B } } ) )
48	47	eqrdv 2620	1 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { p e. ~P { A , B } | p ~~ 2o } = { { A , B } } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   u. cun 3572   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888   omcom 7065   2oc2o 7554   ~~ cen 7952   Fincfn 7955   2c2 11070   #chash 13117

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-pred 5680  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  pmtrprfval  17907