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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 1541 | . . 3 ⊢ ⊤ | |
2 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
3 | rabtru.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nftru 1801 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
5 | biidd 262 | . . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤)) | |
6 | 2, 3, 4, 5 | elrabf 3691 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤)) |
7 | 1, 6 | mpbiran2 710 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴) |
8 | 7 | eqriv 2732 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 Ⅎwnfc 2888 {crab 3433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 |
This theorem is referenced by: mptexgf 7242 aciunf1 32680 |
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