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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 5313 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | eleq1 2827 | . . . . . . 7 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 258 | . . . . . 6 ⊢ (∩ 𝐴 = ∅ → ∩ 𝐴 ∈ V) |
4 | intex 5350 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 234 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → 𝐴 ≠ ∅) |
6 | onint 7810 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | sylan2 593 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∩ 𝐴 ∈ 𝐴) |
8 | eleq1 2827 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) | |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
10 | 7, 9 | mpbid 232 | . . 3 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∅ ∈ 𝐴) |
11 | 10 | ex 412 | . 2 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ → ∅ ∈ 𝐴)) |
12 | int0el 4984 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) | |
13 | 11, 12 | impbid1 225 | 1 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ∩ cint 4951 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: cfeq0 10294 |
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