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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 3494 should be proved from this one, and vex 3495 should be proved from this proposed definition together with vexw 2802, which would remove from vex 3495 dependency on ax-13 2381 (see also comment of vexw 2802). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-df-v | ⊢ V = {𝑥 ∣ ⊤} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2812 | . 2 ⊢ (V = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤})) | |
2 | vex 3495 | . . 3 ⊢ 𝑦 ∈ V | |
3 | tru 1532 | . . . 4 ⊢ ⊤ | |
4 | 3 | vexw 2802 | . . 3 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤} |
5 | 2, 4 | 2th 265 | . 2 ⊢ (𝑦 ∈ V ↔ 𝑦 ∈ {𝑥 ∣ ⊤}) |
6 | 1, 5 | mpgbir 1791 | 1 ⊢ V = {𝑥 ∣ ⊤} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 {cab 2796 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1531 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 |
This theorem is referenced by: (None) |
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