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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-velpwALT | Structured version Visualization version GIF version | ||
| Description: This theorem bj-velpwALT 37054 and the next theorem bj-elpwgALT 37055 are alternate proofs of velpw 4605 and elpwg 4603 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3557 instead of proving first the general case using elab2g 3680 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2177. In other cases, that order is better (e.g., vsnex 5434 proved before snexg 5435). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-velpwALT | ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-pw 4602 | . . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴}) | 
| 3 | abid 2718 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ 𝑥 ⊆ 𝐴) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 {cab 2714 ⊆ wss 3951 𝒫 cpw 4600 | 
| 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-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-pw 4602 | 
| This theorem is referenced by: bj-elpwgALT 37055 | 
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