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Theorem oneqmini 6354
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 4913 . . . . . 6 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
2 ssel 3925 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝐴𝐵𝐴 ∈ On))
3 ssel 3925 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝑥𝐵𝑥 ∈ On))
42, 3anim12d 609 . . . . . . . . . . 11 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5 ontri1 6337 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
64, 5syl6 35 . . . . . . . . . 10 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
76expdimp 453 . . . . . . . . 9 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝑥𝐵 → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
87pm5.74d 272 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
9 con2b 359 . . . . . . . 8 ((𝑥𝐵 → ¬ 𝑥𝐴) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
108, 9bitrdi 286 . . . . . . 7 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐴 → ¬ 𝑥𝐵)))
1110ralbidv2 3166 . . . . . 6 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
121, 11bitrid 282 . . . . 5 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝐴 𝐵 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
1312biimprd 247 . . . 4 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐴 ¬ 𝑥𝐵𝐴 𝐵))
1413expimpd 454 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 𝐵))
15 intss1 4912 . . . . 5 (𝐴𝐵 𝐵𝐴)
1615a1i 11 . . . 4 (𝐵 ⊆ On → (𝐴𝐵 𝐵𝐴))
1716adantrd 492 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐵𝐴))
1814, 17jcad 513 . 2 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → (𝐴 𝐵 𝐵𝐴)))
19 eqss 3947 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
2018, 19syl6ibr 251 1 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  wss 3898   cint 4895  Oncon0 6303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-br 5094  df-opab 5156  df-tr 5211  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-ord 6306  df-on 6307
This theorem is referenced by:  oneqmin  7714  alephval3  9968  cfsuc  10115  alephval2  10430
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