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Mirrors > Home > MPE Home > Th. List > eq0rdvALT | Structured version Visualization version GIF version |
Description: Alternate proof of eq0rdv 4365. Shorter, but requiring df-clel 2811, ax-8 2109. (Contributed by NM, 11-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eq0rdvALT.1 | ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
eq0rdvALT | ⊢ (𝜑 → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0rdvALT.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) | |
2 | 1 | pm2.21d 121 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∅)) |
3 | 2 | ssrdv 3951 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∅) |
4 | ss0 4359 | . 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-nul 4284 |
This theorem is referenced by: (None) |
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