Theorem en2eleq 8831

Description: Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
en2eleq	|- ( ( X e. P /\ P ~~ 2o ) -> P = { X , U. ( P \ { X } ) } )

Proof of Theorem en2eleq

Step	Hyp	Ref	Expression
1		2onn 7720	. . . . . 6 |- 2o e. om
2		nnfi 8153	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
3	1, 2	ax-mp 5	. . . . 5 |- 2o e. Fin
4		enfi 8176	. . . . 5 |- ( P ~~ 2o -> ( P e. Fin <-> 2o e. Fin ) )
5	3, 4	mpbiri 248	. . . 4 |- ( P ~~ 2o -> P e. Fin )
6	5	adantl 482	. . 3 |- ( ( X e. P /\ P ~~ 2o ) -> P e. Fin )
7		simpl 473	. . . 4 |- ( ( X e. P /\ P ~~ 2o ) -> X e. P )
8		1onn 7719	. . . . . . . . 9 |- 1o e. om
9	8	a1i 11	. . . . . . . 8 |- ( ( X e. P /\ P ~~ 2o ) -> 1o e. om )
10		simpr 477	. . . . . . . . 9 |- ( ( X e. P /\ P ~~ 2o ) -> P ~~ 2o )
11		df-2o 7561	. . . . . . . . 9 |- 2o = suc 1o
12	10, 11	syl6breq 4694	. . . . . . . 8 |- ( ( X e. P /\ P ~~ 2o ) -> P ~~ suc 1o )
13		dif1en 8193	. . . . . . . 8 |- ( ( 1o e. om /\ P ~~ suc 1o /\ X e. P ) -> ( P \ { X } ) ~~ 1o )
14	9, 12, 7, 13	syl3anc 1326	. . . . . . 7 |- ( ( X e. P /\ P ~~ 2o ) -> ( P \ { X } ) ~~ 1o )
15		en1uniel 8028	. . . . . . 7 |- ( ( P \ { X } ) ~~ 1o -> U. ( P \ { X } ) e. ( P \ { X } ) )
16	14, 15	syl 17	. . . . . 6 |- ( ( X e. P /\ P ~~ 2o ) -> U. ( P \ { X } ) e. ( P \ { X } ) )
17		eldifsn 4317	. . . . . 6 |- ( U. ( P \ { X } ) e. ( P \ { X } ) <-> ( U. ( P \ { X } ) e. P /\ U. ( P \ { X } ) =/= X ) )
18	16, 17	sylib 208	. . . . 5 |- ( ( X e. P /\ P ~~ 2o ) -> ( U. ( P \ { X } ) e. P /\ U. ( P \ { X } ) =/= X ) )
19	18	simpld 475	. . . 4 |- ( ( X e. P /\ P ~~ 2o ) -> U. ( P \ { X } ) e. P )
20		prssi 4353	. . . 4 |- ( ( X e. P /\ U. ( P \ { X } ) e. P ) -> { X , U. ( P \ { X } ) } C_ P )
21	7, 19, 20	syl2anc 693	. . 3 |- ( ( X e. P /\ P ~~ 2o ) -> { X , U. ( P \ { X } ) } C_ P )
22	18	simprd 479	. . . . . 6 |- ( ( X e. P /\ P ~~ 2o ) -> U. ( P \ { X } ) =/= X )
23	22	necomd 2849	. . . . 5 |- ( ( X e. P /\ P ~~ 2o ) -> X =/= U. ( P \ { X } ) )
24		pr2nelem 8827	. . . . 5 |- ( ( X e. P /\ U. ( P \ { X } ) e. P /\ X =/= U. ( P \ { X } ) ) -> { X , U. ( P \ { X } ) } ~~ 2o )
25	7, 19, 23, 24	syl3anc 1326	. . . 4 |- ( ( X e. P /\ P ~~ 2o ) -> { X , U. ( P \ { X } ) } ~~ 2o )
26		ensym 8005	. . . . 5 |- ( P ~~ 2o -> 2o ~~ P )
27	26	adantl 482	. . . 4 |- ( ( X e. P /\ P ~~ 2o ) -> 2o ~~ P )
28		entr 8008	. . . 4 |- ( ( { X , U. ( P \ { X } ) } ~~ 2o /\ 2o ~~ P ) -> { X , U. ( P \ { X } ) } ~~ P )
29	25, 27, 28	syl2anc 693	. . 3 |- ( ( X e. P /\ P ~~ 2o ) -> { X , U. ( P \ { X } ) } ~~ P )
30		fisseneq 8171	. . 3 |- ( ( P e. Fin /\ { X , U. ( P \ { X } ) } C_ P /\ { X , U. ( P \ { X } ) } ~~ P ) -> { X , U. ( P \ { X } ) } = P )
31	6, 21, 29, 30	syl3anc 1326	. 2 |- ( ( X e. P /\ P ~~ 2o ) -> { X , U. ( P \ { X } ) } = P )
32	31	eqcomd 2628	1 |- ( ( X e. P /\ P ~~ 2o ) -> P = { X , U. ( P \ { X } ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   U.cuni 4436   class class class wbr 4653   suc csuc 5725   omcom 7065   1oc1o 7553   2oc2o 7554   ~~ cen 7952   Fincfn 7955

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

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-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-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-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  en2other2  8832  psgnunilem1  17913