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Mirrors > Home > MPE Home > Th. List > onnmin | Structured version Visualization version GIF version |
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) |
Ref | Expression |
---|---|
onnmin | ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 4890 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ∩ 𝐴 ⊆ 𝐵) |
3 | ne0i 4299 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) | |
4 | oninton 7514 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On) | |
5 | 3, 4 | sylan2 594 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ∩ 𝐴 ∈ On) |
6 | ssel2 3961 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
7 | ontri1 6224 | . . 3 ⊢ ((∩ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (∩ 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ∩ 𝐴)) | |
8 | 5, 6, 7 | syl2anc 586 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → (∩ 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ∩ 𝐴)) |
9 | 2, 8 | mpbid 234 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 ∩ cint 4875 Oncon0 6190 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-tr 5172 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-ord 6193 df-on 6194 |
This theorem is referenced by: onnminsb 7518 oneqmin 7519 onmindif2 7526 cardmin2 9426 ackbij1lem18 9658 cofsmo 9690 fin23lem26 9746 |
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