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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 1544 | . . 3 ⊢ ⊤ | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
| 3 | rabtru.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | nftru 1804 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
| 5 | biidd 262 | . . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤)) | |
| 6 | 2, 3, 4, 5 | elrabf 3688 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤)) |
| 7 | 1, 6 | mpbiran2 710 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴) |
| 8 | 7 | eqriv 2734 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 |
| This theorem is referenced by: mptexgf 7242 aciunf1 32673 |
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