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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrAUTO | Structured version Visualization version GIF version | ||
| Description: Proof of bj-rabtr 36931 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 4080 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
| 2 | ssid 4006 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ 𝐴) | 
| 4 | simpl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⊤) | |
| 5 | 3, 4 | ssrabdv 4074 | . . 3 ⊢ (⊤ → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}) | 
| 6 | 5 | mptru 1547 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} | 
| 7 | 1, 6 | eqssi 4000 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-ss 3968 | 
| This theorem is referenced by: (None) | 
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