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Theorem onnmin 4483
 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.)
Assertion
Ref Expression
onnmin ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Proof of Theorem onnmin
StepHypRef Expression
1 intss1 3786 . . 3 (𝐵𝐴 𝐴𝐵)
2 elirr 4456 . . . 4 ¬ 𝐵𝐵
3 ssel 3091 . . . 4 ( 𝐴𝐵 → (𝐵 𝐴𝐵𝐵))
42, 3mtoi 653 . . 3 ( 𝐴𝐵 → ¬ 𝐵 𝐴)
51, 4syl 14 . 2 (𝐵𝐴 → ¬ 𝐵 𝐴)
65adantl 275 1 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 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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