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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-velpwALT | Structured version Visualization version GIF version |
Description: This theorem bj-velpwALT 37036 and the next theorem bj-elpwgALT 37037 are alternate proofs of velpw 4610 and elpwg 4608 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3557 instead of proving first the general case using elab2g 3683 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2175. In other cases, that order is better (e.g., vsnex 5440 proved before snexg 5441). (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 4607 | . . 3 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝑥 ⊆ 𝐴}) |
3 | abid 2716 | . 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 2106 {cab 2712 ⊆ wss 3963 𝒫 cpw 4605 |
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-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-pw 4607 |
This theorem is referenced by: bj-elpwgALT 37037 |
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