Theorem eq0rdvALT 4365

Description: Alternate proof of eq0rdv 4364. Shorter, but requiring df-clel 2835, ax-8 2146. (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 3915	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∅)
4		ss0 4351	. 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
5	3, 4	syl 18	1 ⊢ (𝜑 → 𝐴 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3897  df-dif 4046  df-nul 4278

This theorem is used by: (None)