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Mirrors > Home > ILE Home > Th. List > bm2.5ii | GIF version |
Description: Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
bm2.5ii.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
bm2.5ii | ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bm2.5ii.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | ssonunii 4506 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
3 | intmin 3879 | . . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴) | |
4 | unissb 3854 | . . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | |
5 | 4 | a1i 9 | . . . . 5 ⊢ (𝑥 ∈ On → (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) |
6 | 5 | rabbiia 2737 | . . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
7 | 6 | inteqi 3863 | . . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
8 | 3, 7 | eqtr3di 2237 | . 2 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
9 | 2, 8 | syl 14 | 1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∀wral 2468 {crab 2472 Vcvv 2752 ⊆ wss 3144 ∪ cuni 3824 ∩ cint 3859 Oncon0 4381 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-in 3150 df-ss 3157 df-uni 3825 df-int 3860 df-tr 4117 df-iord 4384 df-on 4386 |
This theorem is referenced by: (None) |
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