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Theorem onnmin 4228
 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 3604 . . 3 (B A AB)
2 elirr 4208 . . . 4 ¬ B B
3 ssel 2916 . . . 4 ( AB → (B AB B))
42, 3mtoi 577 . . 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 1374   ⊆ wss 2894  ∩ cint 3589  Oncon0 4049 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-sn 3356  df-int 3590 This theorem is referenced by: (None)
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