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Mirrors > Home > MPE Home > Th. List > ab0orv | Structured version Visualization version GIF version |
Description: The class builder of a wff not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) |
Ref | Expression |
---|---|
ab0orv | ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2010 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | dfnf5 4151 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) | |
3 | 1, 2 | mpbi 222 | 1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 874 = wceq 1653 Ⅎwnf 1879 {cab 2783 Vcvv 3383 ∅c0 4113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-v 3385 df-dif 3770 df-nul 4114 |
This theorem is referenced by: (None) |
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