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Mirrors > Home > MPE Home > Th. List > ab0orvALT | Structured version Visualization version GIF version |
Description: Alternate proof of ab0orv 4374, 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 1909 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | dfnf5 4373 | . 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 845 = wceq 1533 Ⅎwnf 1777 {cab 2702 Vcvv 3463 ∅c0 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-v 3465 df-dif 3942 df-nul 4319 |
This theorem is referenced by: (None) |
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