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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtr | Structured version Visualization version GIF version |
Description: Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.) |
Ref | Expression |
---|---|
bj-rabtr | ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4039 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
2 | ssid 3972 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
3 | tru 1541 | . . . 4 ⊢ ⊤ | |
4 | 3 | rgenw 3145 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤ |
5 | ssrab 4032 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤)) | |
6 | 2, 4, 5 | mpbir2an 709 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | 1, 6 | eqssi 3966 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∀wral 3133 {crab 3137 ⊆ wss 3919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rab 3142 df-in 3926 df-ss 3935 |
This theorem is referenced by: (None) |
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