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Theorem onnmin 7493
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 4864 . . 3 (𝐵𝐴 𝐴𝐵)
21adantl 485 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴𝐵)
3 ne0i 4273 . . . 4 (𝐵𝐴𝐴 ≠ ∅)
4 oninton 7490 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
53, 4sylan2 595 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴 ∈ On)
6 ssel2 3938 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
7 ontri1 6198 . . 3 (( 𝐴 ∈ On ∧ 𝐵 ∈ On) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
85, 6, 7syl2anc 587 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
92, 8mpbid 235 1 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2115  wne 3007  wss 3910  c0 4266   cint 4849  Oncon0 6164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-tr 5146  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6167  df-on 6168
This theorem is referenced by:  onnminsb  7494  oneqmin  7495  onmindif2  7502  cardmin2  9404  ackbij1lem18  9636  cofsmo  9668  fin23lem26  9724
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