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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 4215 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
3 | unissb 3610 | . . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝑥 ∈ On → (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) |
5 | 4 | rabbiia 2547 | . . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
6 | 5 | inteqi 3619 | . . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
7 | intmin 3635 | . . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴) | |
8 | 6, 7 | syl5reqr 2087 | . 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 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 {crab 2310 Vcvv 2557 ⊆ wss 2917 ∪ cuni 3580 ∩ cint 3615 Oncon0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-int 3616 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: (None) |
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