Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2151 | . 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) | |
2 | vex 2715 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | tbt 246 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V)) |
4 | 3 | albii 1450 | . 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 1333 = wceq 1335 ∈ wcel 2128 Vcvv 2712 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-v 2714 |
This theorem is referenced by: setindel 4499 dmi 4803 |
Copyright terms: Public domain | W3C validator |