MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oneqmin Structured version   Visualization version   GIF version

Theorem oneqmin 7821
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 7811 . . . 4 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → 𝐵𝐵)
2 eleq1 2828 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 𝐵𝐵))
31, 2syl5ibrcom 247 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵𝐴𝐵))
4 eleq2 2829 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
54biimpd 229 . . . . . 6 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
6 onnmin 7819 . . . . . . . 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 3151 . . . 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 6435 . . 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 1539  wcel 2107  wne 2939  wral 3060  wss 3950  c0 4332   cint 4945  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator