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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-df-v | Structured version Visualization version GIF version |
Description: Alternate definition of the universal class. Actually, the current definition df-v 3175 should be proved from this one, and vex 3176 should be proved from this proposed definition together with bj-vexwv 32051, which would remove from vex 3176 dependency on ax-13 2234 (see also comment of bj-vexw 32049). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-df-v | ⊢ V = {𝑥 ∣ ⊤} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2604 | . 2 ⊢ (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})) | |
2 | vex 3176 | . . 3 ⊢ 𝑦 ∈ V | |
3 | tru 1479 | . . . 4 ⊢ ⊤ | |
4 | 3 | bj-vexwv 32051 | . . 3 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤} |
5 | 2, 4 | 2th 253 | . 2 ⊢ (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) |
6 | 1, 5 | mpgbir 1717 | 1 ⊢ V = {𝑥 ∣ ⊤} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {cab 2596 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-tru 1478 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: (None) |
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