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Mirrors > Home > MPE Home > Th. List > abss | Structured version Visualization version GIF version |
Description: Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
Ref | Expression |
---|---|
abss | ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid2 2882 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
2 | 1 | sseq2i 3950 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
3 | ss2ab 3993 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | bitr3i 276 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∈ wcel 2106 {cab 2715 ⊆ wss 3887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-v 3434 df-in 3894 df-ss 3904 |
This theorem is referenced by: abssdv 4002 rabss 4005 uniiunlem 4019 iunssf 4974 iunss 4975 moabex 5374 reliun 5726 axdc2lem 10204 mptelee 27263 fpwrelmap 31068 ss2iundf 41267 hoidmvlelem1 44133 |
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