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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 2967 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | |
2 | 1 | abbidv 2155 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵}) |
3 | df-pw 3361 | . 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | df-pw 3361 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
5 | 2, 3, 4 | 3eqtr4g 2097 | 1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 {cab 2026 ⊆ wss 2917 𝒫 cpw 3359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-pw 3361 |
This theorem is referenced by: pweqi 3363 pweqd 3364 axpweq 3924 pwex 3932 pwexg 3933 pwssunim 4021 ordpwsucexmid 4294 |
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