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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elpwgALT | Structured version Visualization version GIF version |
Description: Alternate proof of elpwg 4608. See comment for bj-velpwALT 37036. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-elpwgALT | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)) | |
2 | sseq1 4021 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | bj-velpwALT 37036 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) | |
4 | 1, 2, 3 | vtoclbg 3557 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ⊆ 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-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ss 3980 df-pw 4607 |
This theorem is referenced by: (None) |
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