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Mirrors > Home > MPE Home > Th. List > ssabral | Structured version Visualization version GIF version |
Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.) |
Ref | Expression |
---|---|
ssabral | ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssab 3869 | . 2 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | df-ral 3095 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | 1, 2 | bitr4i 270 | 1 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1651 ∈ wcel 2157 {cab 2786 ∀wral 3090 ⊆ wss 3770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-in 3777 df-ss 3784 |
This theorem is referenced by: comppfsc 21663 txdis1cn 21766 qustgplem 22251 xrhmeo 23072 cncmet 23447 itg1addlem4 23806 subfacp1lem6 31683 poimirlem9 33906 istotbnd3 34056 sstotbnd 34060 heibor1lem 34094 heibor1 34095 setis 43238 |
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