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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intabssel1 | GIF version |
Description: Version of intss1 3871 using a class abstraction and implicit substitution. Closed form of intmin3 3883. (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 14803 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
5 | intss1 3871 | . 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 1363 Ⅎwnf 1470 ∈ wcel 2158 {cab 2173 Ⅎwnfc 2316 ⊆ wss 3141 ∩ cint 3856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-in 3147 df-ss 3154 df-int 3857 |
This theorem is referenced by: bj-omssind 14958 |
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