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Mirrors > Home > MPE Home > Th. List > ab0orvALT | Structured version Visualization version GIF version |
Description: Alternate proof of ab0orv 4339, shorter but using more axioms. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ab0orvALT | ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | dfnf5 4338 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) | |
3 | 1, 2 | mpbi 229 | 1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1542 Ⅎwnf 1786 {cab 2710 Vcvv 3444 ∅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-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-v 3446 df-dif 3914 df-nul 4284 |
This theorem is referenced by: (None) |
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