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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 3896 | . 2 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | df-ral 3121 | . 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 1656 ∈ wcel 2166 {cab 2810 ∀wral 3116 ⊆ wss 3797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-in 3804 df-ss 3811 |
This theorem is referenced by: comppfsc 21705 txdis1cn 21808 qustgplem 22293 xrhmeo 23114 cncmet 23489 itg1addlem4 23864 subfacp1lem6 31712 poimirlem9 33961 istotbnd3 34111 sstotbnd 34115 heibor1lem 34149 heibor1 34150 setis 43338 |
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