Theorem eq0rdvALT 4417

Description: Alternate proof of eq0rdv 4416. Shorter, but requiring df-clel 2816, ax-8 2110. (Contributed by NM, 11-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eq0rdvALT.1	⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
eq0rdvALT	⊢ (𝜑 → 𝐴 = ∅)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem eq0rdvALT

Step	Hyp	Ref	Expression
1		eq0rdvALT.1	. . . 4 ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)
2	1	pm2.21d 121	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∅))
3	2	ssrdv 4004	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∅)
4		ss0 4411	. 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
5	3, 4	syl 17	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108   ⊆ wss 3966  ∅c0 4342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3969  df-ss 3983  df-nul 4343

This theorem is referenced by: (None)