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Theorem onnmin 7738
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
Assertion
Ref Expression
onnmin ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Proof of Theorem onnmin
StepHypRef Expression
1 intss1 4929 . . 3 (𝐵𝐴 𝐴𝐵)
21adantl 483 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴𝐵)
3 ne0i 4299 . . . 4 (𝐵𝐴𝐴 ≠ ∅)
4 oninton 7735 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
53, 4sylan2 594 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴 ∈ On)
6 ssel2 3944 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
7 ontri1 6356 . . 3 (( 𝐴 ∈ On ∧ 𝐵 ∈ On) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
85, 6, 7syl2anc 585 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
92, 8mpbid 231 1 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107  wne 2944  wss 3915  c0 4287   cint 4912  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  onnminsb  7739  oneqmin  7740  onmindif2  7747  cardmin2  9942  ackbij1lem18  10180  cofsmo  10212  fin23lem26  10268  lrrecfr  27277
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