Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrALT | Structured version Visualization version GIF version |
Description: Alternate proof of bj-rabtr 32118. (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 3099 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤} | |
2 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | cleqf 2776 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)) |
4 | tru 1479 | . . 3 ⊢ ⊤ | |
5 | rabid 3095 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) | |
6 | 4, 5 | mpbiran2 956 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴) |
7 | 3, 6 | mpgbir 1717 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {crab 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |