Theorem ssequn2 3336

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 3333	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		uncom 3307	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
3	2	eqeq1i 2204	. 2 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
4	1, 3	bitri 184	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1364   ∪ cun 3155   ⊆ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  unabs  3394  pwssunim  4319  pwundifss  4320  oneluni  4466  relresfld  5199  relcoi1  5201  fsnunf  5762  unsnfidcel  6982  tpfidceq  6991  fidcenumlemr  7021  exmidfodomrlemim  7268  ennnfonelemhf1o  12630  lspun0  13981  plyrecj  14999  dvply2g  15002