Theorem dfss5OLD 3781

Description: Obsolete as of 22-Jul-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dfss5OLD	⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴))

Proof of Theorem dfss5OLD

Step	Hyp	Ref	Expression
1		sseqin2 3779	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eqcom 2617	. 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐴 ↔ 𝐴 = (𝐵 ∩ 𝐴))
3	1, 2	bitri 263	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴))

Syntax hints:   ↔ wb 195   = wceq 1475   ∩ cin 3539   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554

This theorem is referenced by: (None)