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Theorem oneqmin 7781
Description: A way to show that an ordinal number equals the minimum of a nonempty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
Assertion
Ref Expression
oneqmin ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oneqmin
StepHypRef Expression
1 onint 7771 . . . 4 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → 𝐵𝐵)
2 eleq1 2813 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 𝐵𝐵))
31, 2syl5ibrcom 246 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵𝐴𝐵))
4 eleq2 2814 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
54biimpd 228 . . . . . 6 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
6 onnmin 7779 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝑥𝐵) → ¬ 𝑥 𝐵)
76ex 412 . . . . . . 7 (𝐵 ⊆ On → (𝑥𝐵 → ¬ 𝑥 𝐵))
87con2d 134 . . . . . 6 (𝐵 ⊆ On → (𝑥 𝐵 → ¬ 𝑥𝐵))
95, 8syl9r 78 . . . . 5 (𝐵 ⊆ On → (𝐴 = 𝐵 → (𝑥𝐴 → ¬ 𝑥𝐵)))
109ralrimdv 3144 . . . 4 (𝐵 ⊆ On → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
1110adantr 480 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
123, 11jcad 512 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
13 oneqmini 6406 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1413adantr 480 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1512, 14impbid 211 1 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wral 3053  wss 3940  c0 4314   cint 4940  Oncon0 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358
This theorem is referenced by: (None)
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