Theorem tpprceq3 4170

Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Assertion

Ref	Expression
tpprceq3

Proof of Theorem tpprceq3

Step	Hyp	Ref	Expression
1		ianor 488	. 2
2		tprot 4125	. . . 4
3		df-tp 4034	. . . . 5
4		prprc2 4141	. . . . . . 7
5	4	uneq1d 3656	. . . . . 6
6		df-pr 4032	. . . . . . 7
7		prcom 4108	. . . . . . 7
8	6, 7	eqtr3i 2488	. . . . . 6
9	5, 8	syl6eq 2514	. . . . 5
10	3, 9	syl5eq 2510	. . . 4
11	2, 10	syl5eq 2510	. . 3
12		nne 2658	. . . 4
13		tppreq3 4135	. . . . 5
14	13	eqcoms 2469	. . . 4
15	12, 14	sylbi 195	. . 3
16	11, 15	jaoi 379	. 2
17	1, 16	sylbi 195	1

Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  cvv 3109  u.cun 3473  {csn 4029  {cpr 4031  {ctp 4033

This theorem is referenced by:  tppreqb  4171  1to3vfriswmgra  25007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032  df-tp 4034