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Mirrors > Home > MPE Home > Th. List > rabtru | Structured version Visualization version GIF version |
Description: Abstract builder using the constant wff ⊤. (Contributed by Thierry Arnoux, 4-May-2020.) |
Ref | Expression |
---|---|
rabtru.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
rabtru | ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1543 | . . 3 ⊢ ⊤ | |
2 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
3 | rabtru.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nftru 1807 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
5 | biidd 261 | . . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤)) | |
6 | 2, 3, 4, 5 | elrabf 3620 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤)) |
7 | 1, 6 | mpbiran2 707 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴) |
8 | 7 | eqriv 2735 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 Ⅎwnfc 2887 {crab 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 |
This theorem is referenced by: mptexgf 7098 aciunf1 31000 |
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