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Theorem oneqmini 6423
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 4968 . . . . . 6 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
2 ssel 3970 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝐴𝐵𝐴 ∈ On))
3 ssel 3970 . . . . . . . . . . . 12 (𝐵 ⊆ On → (𝑥𝐵𝑥 ∈ On))
42, 3anim12d 607 . . . . . . . . . . 11 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5 ontri1 6405 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
64, 5syl6 35 . . . . . . . . . 10 (𝐵 ⊆ On → ((𝐴𝐵𝑥𝐵) → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
76expdimp 451 . . . . . . . . 9 ((𝐵 ⊆ On ∧ 𝐴𝐵) → (𝑥𝐵 → (𝐴𝑥 ↔ ¬ 𝑥𝐴)))
87pm5.74d 272 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐵 → ¬ 𝑥𝐴)))
9 con2b 358 . . . . . . . 8 ((𝑥𝐵 → ¬ 𝑥𝐴) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
108, 9bitrdi 286 . . . . . . 7 ((𝐵 ⊆ On ∧ 𝐴𝐵) → ((𝑥𝐵𝐴𝑥) ↔ (𝑥𝐴 → ¬ 𝑥𝐵)))
1110ralbidv2 3163 . . . . . 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 452 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 𝐵))
15 intss1 4967 . . . . 5 (𝐴𝐵 𝐵𝐴)
1615a1i 11 . . . 4 (𝐵 ⊆ On → (𝐴𝐵 𝐵𝐴))
1716adantrd 490 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐵𝐴))
1814, 17jcad 511 . 2 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → (𝐴 𝐵 𝐵𝐴)))
19 eqss 3992 . 2 (𝐴 = 𝐵 ↔ (𝐴 𝐵 𝐵𝐴))
2018, 19imbitrrdi 251 1 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wss 3944   cint 4950  Oncon0 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375
This theorem is referenced by:  oneqmin  7804  alephval3  10135  cfsuc  10282  alephval2  10597
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