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Mirrors > Home > MPE Home > Th. List > ssintab | Structured version Visualization version GIF version |
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
ssintab | ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4894 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦) | |
2 | sseq2 3995 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | ralab2 3690 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
4 | 1, 3 | bitri 277 | 1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 {cab 2801 ∀wral 3140 ⊆ wss 3938 ∩ cint 4878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-in 3945 df-ss 3954 df-int 4879 |
This theorem is referenced by: ssmin 4897 ssintrab 4901 intmin4 4907 dffi2 8889 rankval3b 9257 sstskm 10266 dfuzi 12076 cycsubg 18353 ssmclslem 32814 mptrcllem 39980 dfrcl2 40026 brtrclfv2 40079 |
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