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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 5308 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | eleq1 2822 | . . . . . . 7 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 258 | . . . . . 6 ⊢ (∩ 𝐴 = ∅ → ∩ 𝐴 ∈ V) |
4 | intex 5338 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 233 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → 𝐴 ≠ ∅) |
6 | onint 7778 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | sylan2 594 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∩ 𝐴 ∈ 𝐴) |
8 | eleq1 2822 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) | |
9 | 8 | adantl 483 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
10 | 7, 9 | mpbid 231 | . . 3 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∅ ∈ 𝐴) |
11 | 10 | ex 414 | . 2 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ → ∅ ∈ 𝐴)) |
12 | int0el 4984 | . 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 ∩ cint 4951 Oncon0 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: cfeq0 10251 |
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