Theorem set2elt 14408

Description: Building a set with two elements.

Assertion

Ref	Expression
set2elt	|- ((C ~~ 2o /\ A e. C /\ B e. C) -> (A =/= B -> C = {A, B}))

Proof of Theorem set2elt

Step	Hyp	Ref	Expression
1		prssg 3140	. . . . 5 |- ((A e. C /\ B e. C) -> ((A e. C /\ B e. C) <-> {A, B} C_ C))
2		2onn 5311	. . . . . . . . . . 11 |- 2o e. om
3		ssid 2634	. . . . . . . . . . 11 |- 2o C_ 2o
4		ssnnfi 5629	. . . . . . . . . . 11 |- ((2o e. om /\ 2o C_ 2o) -> 2o e. Fin)
5	2, 3, 4	mp2an 761	. . . . . . . . . 10 |- 2o e. Fin
6		enfi 5627	. . . . . . . . . . 11 |- ((2o e. om /\ C ~~ 2o) -> (C e. Fin <-> 2o e. Fin))
7	2, 6	mpan 759	. . . . . . . . . 10 |- (C ~~ 2o -> (C e. Fin <-> 2o e. Fin))
8	5, 7	mpbiri 211	. . . . . . . . 9 |- (C ~~ 2o -> C e. Fin)
9		entr 5473	. . . . . . . . . . . . 13 |- ((C ~~ 2o /\ 2o ~~ {A, B}) -> C ~~ {A, B})
10		prex 3526	. . . . . . . . . . . . . . 15 |- {A, B} e. _V
11	10	ensym 5471	. . . . . . . . . . . . . 14 |- (C ~~ {A, B} -> {A, B} ~~ C)
12		sfseqeq 10169	. . . . . . . . . . . . . . . . 17 |- ((C e. Fin /\ {A, B} C_ C /\ {A, B} ~~ C) -> {A, B} = C)
13	12	eqcomd 1889	. . . . . . . . . . . . . . . 16 |- ((C e. Fin /\ {A, B} C_ C /\ {A, B} ~~ C) -> C = {A, B})
14	13	3exp 1066	. . . . . . . . . . . . . . 15 |- (C e. Fin -> ({A, B} C_ C -> ({A, B} ~~ C -> C = {A, B})))
15	14	com3r 39	. . . . . . . . . . . . . 14 |- ({A, B} ~~ C -> (C e. Fin -> ({A, B} C_ C -> C = {A, B})))
16	11, 15	syl 12	. . . . . . . . . . . . 13 |- (C ~~ {A, B} -> (C e. Fin -> ({A, B} C_ C -> C = {A, B})))
17	9, 16	syl 12	. . . . . . . . . . . 12 |- ((C ~~ 2o /\ 2o ~~ {A, B}) -> (C e. Fin -> ({A, B} C_ C -> C = {A, B})))
18	17	ex 402	. . . . . . . . . . 11 |- (C ~~ 2o -> (2o ~~ {A, B} -> (C e. Fin -> ({A, B} C_ C -> C = {A, B}))))
19		ensymg 5470	. . . . . . . . . . . 12 |- (2o e. om -> ({A, B} ~~ 2o -> 2o ~~ {A, B}))
20	2, 19	ax-mp 7	. . . . . . . . . . 11 |- ({A, B} ~~ 2o -> 2o ~~ {A, B})
21	18, 20	syl5com 63	. . . . . . . . . 10 |- ({A, B} ~~ 2o -> (C ~~ 2o -> (C e. Fin -> ({A, B} C_ C -> C = {A, B}))))
22	21	com13 37	. . . . . . . . 9 |- (C e. Fin -> (C ~~ 2o -> ({A, B} ~~ 2o -> ({A, B} C_ C -> C = {A, B}))))
23	8, 22	mpcom 60	. . . . . . . 8 |- (C ~~ 2o -> ({A, B} ~~ 2o -> ({A, B} C_ C -> C = {A, B})))
24		unpde2eg2 14406	. . . . . . . 8 |- ((A e. C /\ B e. C /\ A =/= B) -> {A, B} ~~ 2o)
25	23, 24	syl5com 63	. . . . . . 7 |- ((A e. C /\ B e. C /\ A =/= B) -> (C ~~ 2o -> ({A, B} C_ C -> C = {A, B})))
26	25	3expia 1069	. . . . . 6 |- ((A e. C /\ B e. C) -> (A =/= B -> (C ~~ 2o -> ({A, B} C_ C -> C = {A, B}))))
27	26	com24 41	. . . . 5 |- ((A e. C /\ B e. C) -> ({A, B} C_ C -> (C ~~ 2o -> (A =/= B -> C = {A, B}))))
28	1, 27	sylbid 220	. . . 4 |- ((A e. C /\ B e. C) -> ((A e. C /\ B e. C) -> (C ~~ 2o -> (A =/= B -> C = {A, B}))))
29	28	pm2.43i 78	. . 3 |- ((A e. C /\ B e. C) -> (C ~~ 2o -> (A =/= B -> C = {A, B})))
30	29	com12 14	. 2 |- (C ~~ 2o -> ((A e. C /\ B e. C) -> (A =/= B -> C = {A, B})))
31	30	3impib 1065	1 |- ((C ~~ 2o /\ A e. C /\ B e. C) -> (A =/= B -> C = {A, B}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  {cpr 3045   class class class wbr 3338  omcom 3949  2oc2o 5173   ~~ cen 5423  Fincfn 5426

This theorem is referenced by:  top2ind 14897

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430

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