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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 1546 | . . 3 ⊢ ⊤ | |
2 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
3 | rabtru.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nftru 1811 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
5 | biidd 265 | . . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤)) | |
6 | 2, 3, 4, 5 | elrabf 3589 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤)) |
7 | 1, 6 | mpbiran2 710 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴) |
8 | 7 | eqriv 2736 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 Ⅎwnfc 2880 {crab 3058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-rab 3063 df-v 3402 |
This theorem is referenced by: mptexgf 7008 aciunf1 30588 |
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