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Theorem onnmin 4292
 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 3630 . . 3 (𝐵𝐴 𝐴𝐵)
2 elirr 4266 . . . 4 ¬ 𝐵𝐵
3 ssel 2939 . . . 4 ( 𝐴𝐵 → (𝐵 𝐴𝐵𝐵))
42, 3mtoi 590 . . 3 ( 𝐴𝐵 → ¬ 𝐵 𝐴)
51, 4syl 14 . 2 (𝐵𝐴 → ¬ 𝐵 𝐴)
65adantl 262 1 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393   ⊆ wss 2917  ∩ cint 3615  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-sn 3381  df-int 3616 This theorem is referenced by: (None)
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