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Theorem onnmin 4244
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 B A) → ¬ B A)

Proof of Theorem onnmin
StepHypRef Expression
1 intss1 3621 . . 3 (B A AB)
2 elirr 4224 . . . 4 ¬ B B
3 ssel 2933 . . . 4 ( AB → (B AB B))
42, 3mtoi 589 . . 3 ( AB → ¬ B A)
51, 4syl 14 . 2 (B A → ¬ B A)
65adantl 262 1 ((A ⊆ On B A) → ¬ B A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1390  wss 2911   cint 3606  Oncon0 4066
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-sn 3373  df-int 3607
This theorem is referenced by: (None)
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