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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abv | Structured version Visualization version GIF version |
Description: The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abv | ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1911 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
2 | vexwt 2804 | . . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | alrimih 1824 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) |
4 | eqv 3502 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 |
This theorem is referenced by: curryset 34260 currysetlem3 34263 |
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