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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 4428 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦) | |
2 | sseq2 3590 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | ralab2 3338 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
4 | 1, 3 | bitri 263 | 1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 {cab 2596 ∀wral 2896 ⊆ wss 3540 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-int 4411 |
This theorem is referenced by: ssmin 4431 ssintrab 4435 intmin4 4441 dffi2 8212 rankval3b 8572 sstskm 9543 dfuzi 11344 cycsubg 17445 ssmclslem 30716 mptrcllem 36939 dfrcl2 36985 brtrclfv2 37038 |
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