Theorem hash2pwpr 13258

Description: If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)

Assertion

Ref	Expression
hash2pwpr	|- ( ( ( # ` P ) = 2 /\ P e. ~P { X , Y } ) -> P = { X , Y } )

Proof of Theorem hash2pwpr

Step	Hyp	Ref	Expression
1		pwpr 4430	. . . . 5 |- ~P { X , Y } = ( { (/) , { X } } u. { { Y } , { X , Y } } )
2	1	eleq2i 2693	. . . 4 |- ( P e. ~P { X , Y } <-> P e. ( { (/) , { X } } u. { { Y } , { X , Y } } ) )
3		elun 3753	. . . 4 |- ( P e. ( { (/) , { X } } u. { { Y } , { X , Y } } ) <-> ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) )
4	2, 3	bitri 264	. . 3 |- ( P e. ~P { X , Y } <-> ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) )
5		fveq2 6191	. . . . . . 7 |- ( P = (/) -> ( # ` P ) = ( # ` (/) ) )
6		hash0 13158	. . . . . . . . 9 |- ( # ` (/) ) = 0
7	6	eqeq2i 2634	. . . . . . . 8 |- ( ( # ` P ) = ( # ` (/) ) <-> ( # ` P ) = 0 )
8		eqeq1 2626	. . . . . . . . 9 |- ( ( # ` P ) = 0 -> ( ( # ` P ) = 2 <-> 0 = 2 ) )
9		0ne2 11239	. . . . . . . . . 10 |- 0 =/= 2
10		eqneqall 2805	. . . . . . . . . 10 |- ( 0 = 2 -> ( 0 =/= 2 -> P = { X , Y } ) )
11	9, 10	mpi 20	. . . . . . . . 9 |- ( 0 = 2 -> P = { X , Y } )
12	8, 11	syl6bi 243	. . . . . . . 8 |- ( ( # ` P ) = 0 -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
13	7, 12	sylbi 207	. . . . . . 7 |- ( ( # ` P ) = ( # ` (/) ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
14	5, 13	syl 17	. . . . . 6 |- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
15		hashsng 13159	. . . . . . . 8 |- ( X e. _V -> ( # ` { X } ) = 1 )
16		fveq2 6191	. . . . . . . . . . 11 |- ( { X } = P -> ( # ` { X } ) = ( # ` P ) )
17	16	eqcoms 2630	. . . . . . . . . 10 |- ( P = { X } -> ( # ` { X } ) = ( # ` P ) )
18	17	eqeq1d 2624	. . . . . . . . 9 |- ( P = { X } -> ( ( # ` { X } ) = 1 <-> ( # ` P ) = 1 ) )
19		eqeq1 2626	. . . . . . . . . 10 |- ( ( # ` P ) = 1 -> ( ( # ` P ) = 2 <-> 1 = 2 ) )
20		1ne2 11240	. . . . . . . . . . 11 |- 1 =/= 2
21		eqneqall 2805	. . . . . . . . . . 11 |- ( 1 = 2 -> ( 1 =/= 2 -> P = { X , Y } ) )
22	20, 21	mpi 20	. . . . . . . . . 10 |- ( 1 = 2 -> P = { X , Y } )
23	19, 22	syl6bi 243	. . . . . . . . 9 |- ( ( # ` P ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
24	18, 23	syl6bi 243	. . . . . . . 8 |- ( P = { X } -> ( ( # ` { X } ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
25	15, 24	syl5com 31	. . . . . . 7 |- ( X e. _V -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
26		snprc 4253	. . . . . . . 8 |- ( -. X e. _V <-> { X } = (/) )
27		eqeq2 2633	. . . . . . . . 9 |- ( { X } = (/) -> ( P = { X } <-> P = (/) ) )
28	5, 6	syl6eq 2672	. . . . . . . . . . 11 |- ( P = (/) -> ( # ` P ) = 0 )
29	28	eqeq1d 2624	. . . . . . . . . 10 |- ( P = (/) -> ( ( # ` P ) = 2 <-> 0 = 2 ) )
30	29, 11	syl6bi 243	. . . . . . . . 9 |- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
31	27, 30	syl6bi 243	. . . . . . . 8 |- ( { X } = (/) -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
32	26, 31	sylbi 207	. . . . . . 7 |- ( -. X e. _V -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
33	25, 32	pm2.61i 176	. . . . . 6 |- ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
34	14, 33	jaoi 394	. . . . 5 |- ( ( P = (/) \/ P = { X } ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
35		hashsng 13159	. . . . . . . 8 |- ( Y e. _V -> ( # ` { Y } ) = 1 )
36		fveq2 6191	. . . . . . . . . . 11 |- ( { Y } = P -> ( # ` { Y } ) = ( # ` P ) )
37	36	eqcoms 2630	. . . . . . . . . 10 |- ( P = { Y } -> ( # ` { Y } ) = ( # ` P ) )
38	37	eqeq1d 2624	. . . . . . . . 9 |- ( P = { Y } -> ( ( # ` { Y } ) = 1 <-> ( # ` P ) = 1 ) )
39	38, 23	syl6bi 243	. . . . . . . 8 |- ( P = { Y } -> ( ( # ` { Y } ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
40	35, 39	syl5com 31	. . . . . . 7 |- ( Y e. _V -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
41		snprc 4253	. . . . . . . 8 |- ( -. Y e. _V <-> { Y } = (/) )
42		eqeq2 2633	. . . . . . . . 9 |- ( { Y } = (/) -> ( P = { Y } <-> P = (/) ) )
43	5	eqeq1d 2624	. . . . . . . . . 10 |- ( P = (/) -> ( ( # ` P ) = 2 <-> ( # ` (/) ) = 2 ) )
44	6	eqeq1i 2627	. . . . . . . . . . 11 |- ( ( # ` (/) ) = 2 <-> 0 = 2 )
45	44, 11	sylbi 207	. . . . . . . . . 10 |- ( ( # ` (/) ) = 2 -> P = { X , Y } )
46	43, 45	syl6bi 243	. . . . . . . . 9 |- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
47	42, 46	syl6bi 243	. . . . . . . 8 |- ( { Y } = (/) -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
48	41, 47	sylbi 207	. . . . . . 7 |- ( -. Y e. _V -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) )
49	40, 48	pm2.61i 176	. . . . . 6 |- ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
50		ax-1 6	. . . . . 6 |- ( P = { X , Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
51	49, 50	jaoi 394	. . . . 5 |- ( ( P = { Y } \/ P = { X , Y } ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
52	34, 51	jaoi 394	. . . 4 |- ( ( ( P = (/) \/ P = { X } ) \/ ( P = { Y } \/ P = { X , Y } ) ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) )
53		elpri 4197	. . . . 5 |- ( P e. { (/) , { X } } -> ( P = (/) \/ P = { X } ) )
54		elpri 4197	. . . . 5 |- ( P e. { { Y } , { X , Y } } -> ( P = { Y } \/ P = { X , Y } ) )
55	53, 54	orim12i 538	. . . 4 |- ( ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) -> ( ( P = (/) \/ P = { X } ) \/ ( P = { Y } \/ P = { X , Y } ) ) )
56	52, 55	syl11 33	. . 3 |- ( ( # ` P ) = 2 -> ( ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) -> P = { X , Y } ) )
57	4, 56	syl5bi 232	. 2 |- ( ( # ` P ) = 2 -> ( P e. ~P { X , Y } -> P = { X , Y } ) )
58	57	imp 445	1 |- ( ( ( # ` P ) = 2 /\ P e. ~P { X , Y } ) -> P = { X , Y } )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   ` cfv 5888   0cc0 9936   1c1 9937   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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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:  pr2pwpr  13261