![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 2781 | . . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | alrimih 1825 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) |
4 | eqv 3449 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 |
This theorem is referenced by: curryset 34381 currysetlem3 34384 |
Copyright terms: Public domain | W3C validator |