Theorem eq0rdvALT 4401

Description: Alternate proof of eq0rdv 4400. Shorter, but requiring df-clel 2802, ax-8 2100. (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 3978	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∅)
4		ss0 4394	. 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
5	3, 4	syl 17	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098   ⊆ wss 3939  ∅c0 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-dif 3942  df-ss 3956  df-nul 4319

This theorem is referenced by: (None)