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Mirrors > Home > MPE Home > Th. List > pwssb | Structured version Visualization version GIF version |
Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.) |
Ref | Expression |
---|---|
pwssb | ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 5013 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissb 4861 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
3 | 1, 2 | bitri 276 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wral 3135 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-v 3494 df-in 3940 df-ss 3949 df-pw 4537 df-uni 4831 |
This theorem is referenced by: ustuni 22762 metustfbas 23094 dmvlsiga 31287 1stmbfm 31417 2ndmbfm 31418 dya2iocucvr 31441 gneispace 40362 preimafvsspwdm 43426 |
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