ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onnmin GIF version

Theorem onnmin 4550
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 3844 . . 3 (𝐵𝐴 𝐴𝐵)
2 elirr 4523 . . . 4 ¬ 𝐵𝐵
3 ssel 3141 . . . 4 ( 𝐴𝐵 → (𝐵 𝐴𝐵𝐵))
42, 3mtoi 659 . . 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 2141  wss 3121   cint 3829  Oncon0 4346
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-sn 3587  df-int 3830
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator