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Mirrors > Home > ILE Home > Th. List > eqv | Structured version 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 2017 | . 2 ⊢ (A = V ↔ ∀x(x ∈ A ↔ x ∈ V)) | |
2 | vex 2537 | . . . 4 ⊢ x ∈ V | |
3 | 2 | tbt 236 | . . 3 ⊢ (x ∈ A ↔ (x ∈ A ↔ x ∈ V)) |
4 | 3 | albii 1339 | . 2 ⊢ (∀x x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ V)) |
5 | 1, 4 | bitr4i 176 | 1 ⊢ (A = V ↔ ∀x x ∈ A) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1226 = wceq 1228 ∈ wcel 1375 Vcvv 2534 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1316 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-4 1382 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-ext 2005 |
This theorem depends on definitions: df-bi 110 df-sb 1629 df-clab 2010 df-cleq 2016 df-clel 2019 df-v 2536 |
This theorem is referenced by: setindel 4203 dmi 4475 |
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