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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 2141, ax-11 2157, ax-13 2377. (Revised by BJ, 10-Aug-2022.) |
| Ref | Expression |
|---|---|
| eqv | ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq 2730 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | tbt 369 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
| 4 | 3 | albii 1819 | . 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
| 5 | 1, 4 | bitr4i 278 | 1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 |
| This theorem is referenced by: abvALT 3493 dmi 5932 dmep 5934 dfac10 10178 dfac10c 10179 dfac10b 10180 uniwun 10780 fnsingle 35920 bj-abvALT 36908 ttac 43048 nev 43783 |
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