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Theorem oneqmin 7505
Description: A way to show that an ordinal number equals the minimum of a nonempty 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
oneqmin ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oneqmin
StepHypRef Expression
1 onint 7495 . . . 4 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → 𝐵𝐵)
2 eleq1 2901 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 𝐵𝐵))
31, 2syl5ibrcom 250 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵𝐴𝐵))
4 eleq2 2902 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
54biimpd 232 . . . . . 6 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
6 onnmin 7503 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝑥𝐵) → ¬ 𝑥 𝐵)
76ex 416 . . . . . . 7 (𝐵 ⊆ On → (𝑥𝐵 → ¬ 𝑥 𝐵))
87con2d 136 . . . . . 6 (𝐵 ⊆ On → (𝑥 𝐵 → ¬ 𝑥𝐵))
95, 8syl9r 78 . . . . 5 (𝐵 ⊆ On → (𝐴 = 𝐵 → (𝑥𝐴 → ¬ 𝑥𝐵)))
109ralrimdv 3178 . . . 4 (𝐵 ⊆ On → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
1110adantr 484 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
123, 11jcad 516 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
13 oneqmini 6220 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1413adantr 484 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1512, 14impbid 215 1 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wne 3011  wral 3130  wss 3908  c0 4265   cint 4851  Oncon0 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173
This theorem is referenced by: (None)
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