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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 4090 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴 | |
2 | ssid 4018 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
3 | tru 1541 | . . . 4 ⊢ ⊤ | |
4 | 3 | rgenw 3063 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤ |
5 | ssrab 4083 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤)) | |
6 | 2, 4, 5 | mpbir2an 711 | . 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | 1, 6 | eqssi 4012 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∀wral 3059 {crab 3433 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rab 3434 df-ss 3980 |
This theorem is referenced by: (None) |
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