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

Theorem oneqmini 6245
Description: A way to show that an ordinal number equals the minimum of a 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
oneqmini (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oneqmini
StepHypRef Expression
1 ssint 4895 . . . . . 6 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
2 ssel 3964 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝐴𝐵𝐴 ∈ On))
3 ssel 3964 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝑥𝐵𝑥 ∈ On))
42, 3anim12d 610 . . . . . . . . . . 11 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5 ontri1 6228 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
64, 5syl6 35 . . . . . . . . . 10 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
76expdimp 455 . . . . . . . . 9 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝑥𝐵 → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
87pm5.74d 275 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
9 con2b 362 . . . . . . . 8 ((𝑥𝐵 → ¬ 𝑥𝐴) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
108, 9syl6bb 289 . . . . . . 7 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐴 → ¬ 𝑥𝐵)))
1110ralbidv2 3198 . . . . . 6 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
121, 11syl5bb 285 . . . . 5 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝐴 𝐵 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
1312biimprd 250 . . . 4 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐴 ¬ 𝑥𝐵𝐴 𝐵))
1413expimpd 456 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 𝐵))
15 intss1 4894 . . . . 5 (𝐴𝐵 𝐵𝐴)
1615a1i 11 . . . 4 (𝐵 ⊆ On → (𝐴𝐵 𝐵𝐴))
1716adantrd 494 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐵𝐴))
1814, 17jcad 515 . 2 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → (𝐴 𝐵 𝐵𝐴)))
19 eqss 3985 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
2018, 19syl6ibr 254 1 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  wral 3141  wss 3939   cint 4879  Oncon0 6194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-br 5070  df-opab 5132  df-tr 5176  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-ord 6197  df-on 6198
This theorem is referenced by:  oneqmin  7523  alephval3  9539  cfsuc  9682  alephval2  9997
  Copyright terms: Public domain W3C validator