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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elpwgALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of elpwg 4603. See comment for bj-velpwALT 37054. (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 2829 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)) | |
| 2 | sseq1 4009 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
| 3 | bj-velpwALT 37054 | . 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 2108 ⊆ 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-pw 4602 | 
| This theorem is referenced by: (None) | 
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