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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrAUTO | Structured version Visualization version GIF version |
Description: Proof of bj-rabtr 33749 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 3946 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
2 | ssid 3879 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ 𝐴) |
4 | simpl 475 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⊤) | |
5 | 3, 4 | ssrabdv 3940 | . . 3 ⊢ (⊤ → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}) |
6 | 5 | mptru 1514 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | 1, 6 | eqssi 3874 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 {crab 3092 ⊆ wss 3829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rab 3097 df-in 3836 df-ss 3843 |
This theorem is referenced by: (None) |
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