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Mirrors > Home > MPE Home > Th. List > onint0 | Structured version Visualization version GIF version |
Description: The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
onint0 | ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5297 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | eleq1 2813 | . . . . . . 7 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 258 | . . . . . 6 ⊢ (∩ 𝐴 = ∅ → ∩ 𝐴 ∈ V) |
4 | intex 5327 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 233 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → 𝐴 ≠ ∅) |
6 | onint 7771 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | sylan2 592 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∩ 𝐴 ∈ 𝐴) |
8 | eleq1 2813 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) | |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
10 | 7, 9 | mpbid 231 | . . 3 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∅ ∈ 𝐴) |
11 | 10 | ex 412 | . 2 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ → ∅ ∈ 𝐴)) |
12 | int0el 4973 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) | |
13 | 11, 12 | impbid1 224 | 1 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ⊆ wss 3940 ∅c0 4314 ∩ cint 4940 Oncon0 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-tr 5256 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-ord 6357 df-on 6358 |
This theorem is referenced by: cfeq0 10247 |
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