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Theorem oneqmin 7820
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 7810 . . . 4 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → 𝐵𝐵)
2 eleq1 2827 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 𝐵𝐵))
31, 2syl5ibrcom 247 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵𝐴𝐵))
4 eleq2 2828 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
54biimpd 229 . . . . . 6 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
6 onnmin 7818 . . . . . . . 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 3150 . . . 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 6438 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1413adantr 480 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1512, 14impbid 212 1 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wss 3963  c0 4339   cint 4951  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by: (None)
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