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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 | trud 1543 | . . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤) | |
2 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑) | |
3 | 1, 2 | impbida 798 | . . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤)) |
4 | 3 | alimi 1805 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤)) |
5 | abbi 2794 | . . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤}) |
7 | dfv2 3471 | . 2 ⊢ V = {𝑥 ∣ ⊤} | |
8 | 6, 7 | eqtr4di 2784 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ⊤wtru 1534 {cab 2703 Vcvv 3468 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-v 3470 |
This theorem is referenced by: curryset 36334 currysetlem3 36337 |
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