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Theorem oneqmin 6874
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 6864 . . . 4 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → 𝐵𝐵)
2 eleq1 2675 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵 𝐵𝐵))
31, 2syl5ibrcom 235 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵𝐴𝐵))
4 eleq2 2676 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
54biimpd 217 . . . . . 6 (𝐴 = 𝐵 → (𝑥𝐴𝑥 𝐵))
6 onnmin 6872 . . . . . . . 8 ((𝐵 ⊆ On ∧ 𝑥𝐵) → ¬ 𝑥 𝐵)
76ex 448 . . . . . . 7 (𝐵 ⊆ On → (𝑥𝐵 → ¬ 𝑥 𝐵))
87con2d 127 . . . . . 6 (𝐵 ⊆ On → (𝑥 𝐵 → ¬ 𝑥𝐵))
95, 8syl9r 75 . . . . 5 (𝐵 ⊆ On → (𝐴 = 𝐵 → (𝑥𝐴 → ¬ 𝑥𝐵)))
109ralrimdv 2950 . . . 4 (𝐵 ⊆ On → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
1110adantr 479 . . 3 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → ∀𝑥𝐴 ¬ 𝑥𝐵))
123, 11jcad 553 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 → (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
13 oneqmini 5679 . . 3 (𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1413adantr 479 . 2 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
1512, 14impbid 200 1 ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wss 3539  c0 3873   cint 4404  Oncon0 5626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630
This theorem is referenced by: (None)
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