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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrAUTO | Structured version Visualization version GIF version |
Description: Proof of bj-rabtr 34145 found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE_WITH *". (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 4053 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
2 | ssid 3986 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ 𝐴) |
4 | simpl 483 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⊤) | |
5 | 3, 4 | ssrabdv 4047 | . . 3 ⊢ (⊤ → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}) |
6 | 5 | mptru 1535 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | 1, 6 | eqssi 3980 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-in 3940 df-ss 3949 |
This theorem is referenced by: (None) |
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