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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrALT | Structured version Visualization version GIF version |
Description: Alternate proof of bj-rabtr 33419. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-rabtrALT | ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfrab1 3305 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤} | |
2 | nfcv 2942 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | cleqf 2968 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)) |
4 | tru 1658 | . . 3 ⊢ ⊤ | |
5 | rabid 3298 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) | |
6 | 4, 5 | mpbiran2 702 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴) |
7 | 3, 6 | mpgbir 1895 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 {crab 3094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-rab 3099 |
This theorem is referenced by: (None) |
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