Theorem unielrel 5537

Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
unielrel

Proof of Theorem unielrel

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrel 5110	. 2
2		simpr 461	. 2
3		vex 3112	. . . . . 6
4		vex 3112	. . . . . 6
5	3, 4	uniopel 4756	. . . . 5
6	5	a1i 11	. . . 4
7		eleq1 2529	. . . 4
8		unieq 4257	. . . . 5
9	8	eleq1d 2526	. . . 4
10	6, 7, 9	3imtr4d 268	. . 3
11	10	exlimivv 1723	. 2
12	1, 2, 11	sylc 60	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035  U.cuni 4249  Relwrel 5009

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  df-opab 4511  df-xp 5010  df-rel 5011