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Mirrors > Home > MPE Home > Th. List > eqvf | Structured version Visualization version GIF version |
Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021.) |
Ref | Expression |
---|---|
eqvf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
eqvf | ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑥V | |
3 | 1, 2 | cleqf 3010 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
4 | vex 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | tbt 372 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
6 | 5 | albii 1816 | . 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
7 | 3, 6 | bitr4i 280 | 1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 |
This theorem is referenced by: (None) |
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