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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 1553 | . . . . 5 ⊢ ((𝜑 ∧ 𝜑) → ⊤) | |
2 | simpl 486 | . . . . 5 ⊢ ((𝜑 ∧ ⊤) → 𝜑) | |
3 | 1, 2 | impbida 801 | . . . 4 ⊢ (𝜑 → (𝜑 ↔ ⊤)) |
4 | 3 | alimi 1819 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ↔ ⊤)) |
5 | abbi1 2808 | . . 3 ⊢ (∀𝑥(𝜑 ↔ ⊤) → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤}) |
7 | dfv2 3425 | . 2 ⊢ V = {𝑥 ∣ ⊤} | |
8 | 6, 7 | eqtr4di 2798 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ⊤wtru 1544 {cab 2716 Vcvv 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-v 3424 |
This theorem is referenced by: curryset 34903 currysetlem3 34906 |
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