Theorem ssequn2 3300

Description: A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Assertion

Ref	Expression
ssequn2	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)

Proof of Theorem ssequn2

Step	Hyp	Ref	Expression
1		ssequn1 3297	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		uncom 3271	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
3	2	eqeq1i 2178	. 2 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
4	1, 3	bitri 183	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)

Syntax hints:   ↔ wb 104   = wceq 1348   ∪ cun 3119   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  unabs  3358  pwssunim  4269  pwundifss  4270  oneluni  4416  relresfld  5140  relcoi1  5142  fsnunf  5696  unsnfidcel  6898  fidcenumlemr  6932  exmidfodomrlemim  7178  ennnfonelemhf1o  12368