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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrAUTO | Structured version Visualization version GIF version |
Description: Proof of bj-rabtr 32118 found automatically by "improve all /depth 3 /3" followed by "minimize *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-rabtrAUTO | ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3650 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
2 | ssid 3587 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ 𝐴) |
4 | simpl 472 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⊤) | |
5 | 3, 4 | ssrabdv 3644 | . . 3 ⊢ (⊤ → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}) |
6 | 5 | trud 1484 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | 1, 6 | eqssi 3584 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 |
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-ral 2901 df-rab 2905 df-in 3547 df-ss 3554 |
This theorem is referenced by: (None) |
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