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Mirrors > Home > MPE Home > Th. List > eqv | Structured version Visualization version GIF version |
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2136, ax-11 2151, ax-13 2381. (Revised by BJ, 10-Aug-2022.) |
Ref | Expression |
---|---|
eqv | ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2812 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) | |
2 | vex 3495 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | tbt 371 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
4 | 3 | albii 1811 | . 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
5 | 1, 4 | bitr4i 279 | 1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 |
This theorem is referenced by: abv 3502 dmi 5784 dmep 5786 dfac10 9551 dfac10c 9552 dfac10b 9553 uniwun 10150 fnsingle 33277 bj-abv 34120 ttac 39511 nev 39993 |
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