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Mirrors > Home > ILE Home > Th. List > ssintab | 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 3757 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦) | |
2 | sseq2 3091 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | ralab2 2821 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ⊆ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
4 | 1, 3 | bitri 183 | 1 ⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 {cab 2103 ∀wral 2393 ⊆ wss 3041 ∩ cint 3741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 |
This theorem is referenced by: ssmin 3760 ssintrab 3764 intmin4 3769 dfuzi 9129 |
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