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Theorem oneqmini 5693
 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 4428 . . . . . 6 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
2 ssel 3562 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝐴𝐵𝐴 ∈ On))
3 ssel 3562 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝑥𝐵𝑥 ∈ On))
42, 3anim12d 584 . . . . . . . . . . 11 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5 ontri1 5674 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
64, 5syl6 34 . . . . . . . . . 10 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
76expdimp 452 . . . . . . . . 9 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝑥𝐵 → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
87pm5.74d 261 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
9 con2b 348 . . . . . . . 8 ((𝑥𝐵 → ¬ 𝑥𝐴) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
108, 9syl6bb 275 . . . . . . 7 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐴 → ¬ 𝑥𝐵)))
1110ralbidv2 2967 . . . . . 6 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
121, 11syl5bb 271 . . . . 5 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝐴 𝐵 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
1312biimprd 237 . . . 4 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐴 ¬ 𝑥𝐵𝐴 𝐵))
1413expimpd 627 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 𝐵))
15 intss1 4427 . . . . 5 (𝐴𝐵 𝐵𝐴)
1615a1i 11 . . . 4 (𝐵 ⊆ On → (𝐴𝐵 𝐵𝐴))
1716adantrd 483 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐵𝐴))
1814, 17jcad 554 . 2 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → (𝐴 𝐵 𝐵𝐴)))
19 eqss 3583 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
2018, 19syl6ibr 241 1 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∩ cint 4410  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  oneqmin  6897  alephval3  8816  cfsuc  8962  alephval2  9273
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