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Mirrors > Home > ILE Home > Th. List > intmin3 | GIF version |
Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.) |
Ref | Expression |
---|---|
intmin3.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
intmin3.3 | ⊢ 𝜓 |
Ref | Expression |
---|---|
intmin3 | ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intmin3.3 | . . 3 ⊢ 𝜓 | |
2 | intmin3.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | elabg 2799 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | mpbiri 167 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
5 | intss1 3752 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ∈ wcel 1463 {cab 2101 ⊆ wss 3037 ∩ cint 3737 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-in 3043 df-ss 3050 df-int 3738 |
This theorem is referenced by: intid 4106 |
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