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Mirrors > Home > MPE Home > Th. List > uniordint | Structured version Visualization version GIF version |
Description: The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
uniordint.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
uniordint | ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniordint.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | ssonunii 7324 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) |
3 | unissb 4748 | . . . . 5 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | |
4 | 3 | rabbii 3401 | . . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
5 | 4 | inteqi 4758 | . . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} |
6 | intmin 4774 | . . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴) | |
7 | 5, 6 | syl5reqr 2831 | . 2 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
8 | 2, 7 | syl 17 | 1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ∀wral 3090 {crab 3094 Vcvv 3417 ⊆ wss 3831 ∪ cuni 4717 ∩ cint 4754 Oncon0 6034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3419 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-br 4935 df-opab 4997 df-tr 5036 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-ord 6037 df-on 6038 |
This theorem is referenced by: (None) |
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