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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intabssel1 | GIF version |
Description: Version of intss1 3709 using a class abstraction and implicit substitution. Closed form of intmin3 3721. (Contributed by BJ, 29-Nov-2019.) |
Ref | Expression |
---|---|
bj-intabssel1.nf | ⊢ Ⅎ𝑥𝐴 |
bj-intabssel1.nf2 | ⊢ Ⅎ𝑥𝜓 |
bj-intabssel1.is | ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) |
Ref | Expression |
---|---|
bj-intabssel1 | ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-intabssel1.nf | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | bj-intabssel1.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-intabssel1.is | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
4 | 1, 2, 3 | elabgf2 11953 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
5 | intss1 3709 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl6 33 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 Ⅎwnf 1395 ∈ wcel 1439 {cab 2075 Ⅎwnfc 2216 ⊆ wss 3000 ∩ cint 3694 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-int 3695 |
This theorem is referenced by: bj-omssind 12103 |
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