Theorem uniop 4755

Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
opthw.1
opthw.2

Assertion

Ref	Expression
uniop

Proof of Theorem uniop

Step	Hyp	Ref	Expression
1		opthw.1	. . . 4
2		opthw.2	. . . 4
3	1, 2	dfop 4216	. . 3
4	3	unieqi 4258	. 2
5		snex 4693	. . 3
6		prex 4694	. . 3
7	5, 6	unipr 4262	. 2
8		snsspr1 4179	. . 3
9		ssequn1 3673	. . 3
10	8, 9	mpbi 208	. 2
11	4, 7, 10	3eqtri 2490	1

Syntax hints:  =wceq 1395  e.wcel 1818  cvv 3109  u.cun 3473  C_wss 3475  {csn 4029  {cpr 4031  <.cop 4035  U.cuni 4249

This theorem is referenced by:  uniopel  4756  elvvuni  5065  dmrnssfld  5266  dffv2  5946  rankxplim  8318

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  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-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250