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Mirrors > Home > MPE Home > Th. List > abv | Structured version Visualization version GIF version |
Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 36276) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2802, ax-8 2100. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.) |
Ref | Expression |
---|---|
abv | ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2717 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤})) | |
2 | vextru 2708 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤} | |
3 | 2 | tbt 369 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤})) |
4 | df-clab 2702 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
5 | 3, 4 | bitr3i 277 | . . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ [𝑦 / 𝑥]𝜑) |
6 | 5 | albii 1813 | . . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
7 | 1, 6 | bitri 275 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
8 | dfv2 3469 | . . 3 ⊢ V = {𝑥 ∣ ⊤} | |
9 | 8 | eqeq2i 2737 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤}) |
10 | sb8v 2340 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | |
11 | 7, 9, 10 | 3bitr4i 303 | 1 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1531 = wceq 1533 ⊤wtru 1534 [wsb 2059 ∈ wcel 2098 {cab 2701 Vcvv 3466 |
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-11 2146 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-v 3468 |
This theorem is referenced by: dfnf5 4369 |
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