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Mirrors > Home > ILE Home > Th. List > pweq | GIF version |
Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
pweq | ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3087 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | |
2 | 1 | abbidv 2232 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵}) |
3 | df-pw 3478 | . 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | df-pw 3478 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
5 | 2, 3, 4 | 3eqtr4g 2172 | 1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 {cab 2101 ⊆ wss 3037 𝒫 cpw 3476 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-11 1467 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-in 3043 df-ss 3050 df-pw 3478 |
This theorem is referenced by: pweqi 3480 pweqd 3481 axpweq 4055 pwexg 4064 pwssunim 4166 ordpwsucexmid 4445 fival 6810 istopg 12009 istopon 12023 eltg 12064 tgdom 12084 ntrval 12122 |
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