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Theorem onssmin 6982
Description: A nonempty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)
Assertion
Ref Expression
onssmin ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem onssmin
StepHypRef Expression
1 onint 6980 . 2 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
2 intss1 4483 . . 3 (𝑦𝐴 𝐴𝑦)
32rgen 2919 . 2 𝑦𝐴 𝐴𝑦
4 sseq1 3618 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦 𝐴𝑦))
54ralbidv 2983 . . 3 (𝑥 = 𝐴 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 𝐴𝑦))
65rspcev 3304 . 2 (( 𝐴𝐴 ∧ ∀𝑦𝐴 𝐴𝑦) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
71, 3, 6sylancl 693 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  wss 3567  c0 3907   cint 4466  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715
This theorem is referenced by: (None)
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