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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 5217 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | eleq1 2827 | . . . . . . 7 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 261 | . . . . . 6 ⊢ (∩ 𝐴 = ∅ → ∩ 𝐴 ∈ V) |
4 | intex 5247 | . . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 237 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → 𝐴 ≠ ∅) |
6 | onint 7596 | . . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
7 | 5, 6 | sylan2 596 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∩ 𝐴 ∈ 𝐴) |
8 | eleq1 2827 | . . . . 5 ⊢ (∩ 𝐴 = ∅ → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) | |
9 | 8 | adantl 485 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → (∩ 𝐴 ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
10 | 7, 9 | mpbid 235 | . . 3 ⊢ ((𝐴 ⊆ On ∧ ∩ 𝐴 = ∅) → ∅ ∈ 𝐴) |
11 | 10 | ex 416 | . 2 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ → ∅ ∈ 𝐴)) |
12 | int0el 4907 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) | |
13 | 11, 12 | impbid1 228 | 1 ⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2943 Vcvv 3423 ⊆ wss 3883 ∅c0 4254 ∩ cint 4876 Oncon0 6234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 ax-un 7545 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-br 5071 df-opab 5133 df-tr 5179 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-ord 6237 df-on 6238 |
This theorem is referenced by: cfeq0 9900 |
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