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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 2892 | . . 3 ⊢ Ⅎ𝑥V | |
3 | 1, 2 | cleqf 2924 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
4 | vex 3466 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | tbt 368 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
6 | 5 | albii 1814 | . 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
7 | 3, 6 | bitr4i 277 | 1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1532 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2876 Vcvv 3462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-v 3464 |
This theorem is referenced by: (None) |
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