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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabtrALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of bj-rabtr 36931. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-rabtrALT | ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfrab1 3457 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤} | |
| 2 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 1, 2 | cleqf 2934 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)) | 
| 4 | tru 1544 | . . 3 ⊢ ⊤ | |
| 5 | rabid 3458 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) | |
| 6 | 4, 5 | mpbiran2 710 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴) | 
| 7 | 3, 6 | mpgbir 1799 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 {crab 3436 | 
| 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-rab 3437 | 
| This theorem is referenced by: (None) | 
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