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Theorem onnmin 7788
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 4967 . . 3 (𝐵𝐴 𝐴𝐵)
21adantl 482 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴𝐵)
3 ne0i 4334 . . . 4 (𝐵𝐴𝐴 ≠ ∅)
4 oninton 7785 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
53, 4sylan2 593 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴 ∈ On)
6 ssel2 3977 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
7 ontri1 6398 . . 3 (( 𝐴 ∈ On ∧ 𝐵 ∈ On) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
85, 6, 7syl2anc 584 . 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 396  wcel 2106  wne 2940  wss 3948  c0 4322   cint 4950  Oncon0 6364
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  onnminsb  7789  oneqmin  7790  onmindif2  7797  cardmin2  9996  ackbij1lem18  10234  cofsmo  10266  fin23lem26  10322  lrrecfr  27653
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