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Theorem oneqmini 6435
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 4963 . . . . . 6 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
2 ssel 3976 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝐴𝐵𝐴 ∈ On))
3 ssel 3976 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝑥𝐵𝑥 ∈ On))
42, 3anim12d 609 . . . . . . . . . . 11 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5 ontri1 6417 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
64, 5syl6 35 . . . . . . . . . 10 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
76expdimp 452 . . . . . . . . 9 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝑥𝐵 → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
87pm5.74d 273 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
9 con2b 359 . . . . . . . 8 ((𝑥𝐵 → ¬ 𝑥𝐴) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
108, 9bitrdi 287 . . . . . . 7 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐴 → ¬ 𝑥𝐵)))
1110ralbidv2 3173 . . . . . 6 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
121, 11bitrid 283 . . . . 5 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝐴 𝐵 ↔ ∀𝑥𝐴 ¬ 𝑥𝐵))
1312biimprd 248 . . . 4 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (∀𝑥𝐴 ¬ 𝑥𝐵𝐴 𝐵))
1413expimpd 453 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 𝐵))
15 intss1 4962 . . . . 5 (𝐴𝐵 𝐵𝐴)
1615a1i 11 . . . 4 (𝐵 ⊆ On → (𝐴𝐵 𝐵𝐴))
1716adantrd 491 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐵𝐴))
1814, 17jcad 512 . 2 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → (𝐴 𝐵 𝐵𝐴)))
19 eqss 3998 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
2018, 19imbitrrdi 252 1 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wss 3950   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:  oneqmin  7821  alephval3  10151  cfsuc  10298  alephval2  10613
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