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Mirrors > Home > MPE Home > Th. List > prelpw | Structured version Visualization version GIF version |
Description: A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.) |
Ref | Expression |
---|---|
prelpw | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssg 4712 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)) | |
2 | prex 5298 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
3 | 2 | elpw 4501 | . 2 ⊢ ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶) |
4 | 1, 3 | syl6bbr 292 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 {cpr 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 |
This theorem is referenced by: prelpwi 5305 hashle2prv 13832 umgrpredgv 26933 prprelb 44033 |
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