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Theorem onnmin 7168
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 4644 . . 3 (𝐵𝐴 𝐴𝐵)
21adantl 473 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴𝐵)
3 ne0i 4064 . . . 4 (𝐵𝐴𝐴 ≠ ∅)
4 oninton 7165 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
53, 4sylan2 492 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐴 ∈ On)
6 ssel2 3739 . . 3 ((𝐴 ⊆ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
7 ontri1 5918 . . 3 (( 𝐴 ∈ On ∧ 𝐵 ∈ On) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
85, 6, 7syl2anc 696 . 2 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ( 𝐴𝐵 ↔ ¬ 𝐵 𝐴))
92, 8mpbid 222 1 ((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2139  wne 2932  wss 3715  c0 4058   cint 4627  Oncon0 5884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888
This theorem is referenced by:  onnminsb  7169  oneqmin  7170  onmindif2  7177  cardmin2  9014  ackbij1lem18  9251  cofsmo  9283  fin23lem26  9339
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