Theorem dfss2OLD 3908

Description: Obsolete version of dfss2 3907 as of 16-May-2024. (Contributed by NM, 8-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfss2OLD	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss2OLD

Step	Hyp	Ref	Expression
1		dfss 3905	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
2		df-in 3894	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
3	2	eqeq2i 2751	. . 3 ⊢ (𝐴 = (𝐴 ∩ 𝐵) ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)})
4		abeq2 2872	. . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
5	1, 3, 4	3bitri 297	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
6		pm4.71 558	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
7	6	albii 1822	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
8	5, 7	bitr4i 277	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715   ∩ cin 3886   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-in 3894  df-ss 3904

This theorem is referenced by: (None)