Theorem ssequn2 3310

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 3307	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		uncom 3281	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
3	2	eqeq1i 2185	. 2 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
4	1, 3	bitri 184	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1353   ∪ cun 3129   ⊆ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144

This theorem is referenced by:  unabs  3368  pwssunim  4286  pwundifss  4287  oneluni  4433  relresfld  5160  relcoi1  5162  fsnunf  5719  unsnfidcel  6923  fidcenumlemr  6957  exmidfodomrlemim  7203  ennnfonelemhf1o  12417  lspun0  13517