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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 4460 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
3 | intmin 3838 | . . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴) | |
4 | unissb 3813 | . . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | |
5 | 4 | a1i 9 | . . . . 5 ⊢ (𝑥 ∈ On → (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) |
6 | 5 | rabbiia 2706 | . . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
7 | 6 | inteqi 3822 | . . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
8 | 3, 7 | eqtr3di 2212 | . 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 104 = wceq 1342 ∈ wcel 2135 ∀wral 2442 {crab 2446 Vcvv 2721 ⊆ wss 3111 ∪ cuni 3783 ∩ cint 3818 Oncon0 4335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-in 3117 df-ss 3124 df-uni 3784 df-int 3819 df-tr 4075 df-iord 4338 df-on 4340 |
This theorem is referenced by: (None) |
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