Theorem eq0rdvALT 4369

Description: Alternate proof of eq0rdv 4368. Shorter, but requiring df-clel 2839, ax-8 2149. (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 122	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∅))
3	2	ssrdv 3919	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∅)
4		ss0 4355	. 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
5	3, 4	syl 18	1 ⊢ (𝜑 → 𝐴 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-dif 4050  df-nul 4282

This theorem is used by: (None)