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Mirrors > Home > ILE Home > Th. List > onnmin | GIF version |
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
onnmin | ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3786 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝐵) | |
2 | elirr 4456 | . . . 4 ⊢ ¬ 𝐵 ∈ 𝐵 | |
3 | ssel 3091 | . . . 4 ⊢ (∩ 𝐴 ⊆ 𝐵 → (𝐵 ∈ ∩ 𝐴 → 𝐵 ∈ 𝐵)) | |
4 | 2, 3 | mtoi 653 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ ∩ 𝐴) |
5 | 1, 4 | syl 14 | . 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ ∩ 𝐴) |
6 | 5 | adantl 275 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1480 ⊆ wss 3071 ∩ cint 3771 Oncon0 4285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-sn 3533 df-int 3772 |
This theorem is referenced by: (None) |
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