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Mirrors > Home > ILE Home > Th. List > eqv | GIF version |
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) |
Ref | Expression |
---|---|
eqv | ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2164 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) | |
2 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | tbt 246 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
4 | 3 | albii 1463 | . 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
5 | 1, 4 | bitr4i 186 | 1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1346 = wceq 1348 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: setindel 4522 dmi 4826 |
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