Theorem uniprg 2516

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.

Assertion

Ref	Expression
uniprg	|- ((A e. C /\ B e. D) -> U.{A, B} = (A u. B))

Proof of Theorem uniprg

Step	Hyp	Ref	Expression
1		preq1 2448	. . . 4 |- (x = A -> {x, y} = {A, y})
2	1	unieqd 2512	. . 3 |- (x = A -> U.{x, y} = U.{A, y})
3		uneq1 2177	. . 3 |- (x = A -> (x u. y) = (A u. y))
4	2, 3	eqeq12d 1489	. 2 |- (x = A -> (U.{x, y} = (x u. y) <-> U.{A, y} = (A u. y)))
5		preq2 2449	. . . 4 |- (y = B -> {A, y} = {A, B})
6	5	unieqd 2512	. . 3 |- (y = B -> U.{A, y} = U.{A, B})
7		uneq2 2178	. . 3 |- (y = B -> (A u. y) = (A u. B))
8	6, 7	eqeq12d 1489	. 2 |- (y = B -> (U.{A, y} = (A u. y) <-> U.{A, B} = (A u. B)))
9		visset 1813	. . 3 |- x e. V
10		visset 1813	. . 3 |- y e. V
11	9, 10	unipr 2515	. 2 |- U.{x, y} = (x u. y)
12	4, 8, 11	vtocl2g 1850	1 |- ((A e. C /\ B e. D) -> U.{A, B} = (A u. B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   u. cun 2045  {cpr 2410  U.cuni 2503

This theorem is referenced by:  unctb 7577  cdrci 10494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-uni 2504

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