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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 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | eleq1 2676 | . . . . . . 7 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 247 | . . . . . 6 ⊢ (∩ 𝐴 = ∅ → ∩ 𝐴 ∈ V) |
4 | intex 4747 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 223 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → 𝐴 ≠ ∅) |
6 | onint 6887 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | sylan2 490 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∩ 𝐴 ∈ 𝐴) |
8 | eleq1 2676 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) | |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
10 | 7, 9 | mpbid 221 | . . 3 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∅ ∈ 𝐴) |
11 | 10 | ex 449 | . 2 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ → ∅ ∈ 𝐴)) |
12 | int0el 4443 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) | |
13 | 11, 12 | impbid1 214 | 1 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∩ cint 4410 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: cfeq0 8961 |
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