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Theorem eq0rdvALT 4320
Description: Alternate proof of eq0rdv 4319. Shorter, but requiring df-clel 2816, ax-8 2112. (Contributed by NM, 11-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eq0rdvALT.1 (𝜑 → ¬ 𝑥𝐴)
Assertion
Ref Expression
eq0rdvALT (𝜑𝐴 = ∅)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem eq0rdvALT
StepHypRef Expression
1 eq0rdvALT.1 . . . 4 (𝜑 → ¬ 𝑥𝐴)
21pm2.21d 121 . . 3 (𝜑 → (𝑥𝐴𝑥 ∈ ∅))
32ssrdv 3907 . 2 (𝜑𝐴 ⊆ ∅)
4 ss0 4313 . 2 (𝐴 ⊆ ∅ → 𝐴 = ∅)
53, 4syl 17 1 (𝜑𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2110  wss 3866  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238
This theorem is referenced by: (None)
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