Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2977 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
3 | rabtru.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nftru 1805 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
5 | biidd 264 | . . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤)) | |
6 | 2, 3, 4, 5 | elrabf 3676 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤)) |
7 | 1, 6 | mpbiran2 708 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴) |
8 | 7 | eqriv 2818 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 Ⅎwnfc 2961 {crab 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 |
This theorem is referenced by: mptexgf 6985 aciunf1 30408 |
Copyright terms: Public domain | W3C validator |