Theorem eq0rdvALT 4362

Description: Alternate proof of eq0rdv 4361. Shorter, but requiring df-clel 2812, ax-8 2116. (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 3941	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∅)
4		ss0 4356	. 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
5	3, 4	syl 17	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-dif 3906  df-ss 3920  df-nul 4288

This theorem is referenced by: (None)