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Theorem ord1eln01 8446
Description: An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
Assertion
Ref Expression
ord1eln01 (Ord 𝐴 → (1o𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))

Proof of Theorem ord1eln01
StepHypRef Expression
1 ne0i 4298 . . 3 (1o𝐴𝐴 ≠ ∅)
2 1on 8428 . . . . . 6 1o ∈ On
32onirri 6434 . . . . 5 ¬ 1o ∈ 1o
4 eleq2 2823 . . . . 5 (𝐴 = 1o → (1o𝐴 ↔ 1o ∈ 1o))
53, 4mtbiri 327 . . . 4 (𝐴 = 1o → ¬ 1o𝐴)
65necon2ai 2970 . . 3 (1o𝐴𝐴 ≠ 1o)
71, 6jca 513 . 2 (1o𝐴 → (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o))
8 el1o 8445 . . . . . . 7 (𝐴 ∈ 1o𝐴 = ∅)
98biimpi 215 . . . . . 6 (𝐴 ∈ 1o𝐴 = ∅)
109necon3ai 2965 . . . . 5 (𝐴 ≠ ∅ → ¬ 𝐴 ∈ 1o)
11 nesym 2997 . . . . . 6 (𝐴 ≠ 1o ↔ ¬ 1o = 𝐴)
1211biimpi 215 . . . . 5 (𝐴 ≠ 1o → ¬ 1o = 𝐴)
1310, 12anim12ci 615 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → (¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o))
14 pm4.56 988 . . . 4 ((¬ 1o = 𝐴 ∧ ¬ 𝐴 ∈ 1o) ↔ ¬ (1o = 𝐴𝐴 ∈ 1o))
1513, 14sylib 217 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → ¬ (1o = 𝐴𝐴 ∈ 1o))
162onordi 6432 . . . 4 Ord 1o
17 ordtri2 6356 . . . 4 ((Ord 1o ∧ Ord 𝐴) → (1o𝐴 ↔ ¬ (1o = 𝐴𝐴 ∈ 1o)))
1816, 17mpan 689 . . 3 (Ord 𝐴 → (1o𝐴 ↔ ¬ (1o = 𝐴𝐴 ∈ 1o)))
1915, 18imbitrrid 245 . 2 (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o) → 1o𝐴))
207, 19impbid2 225 1 (Ord 𝐴 → (1o𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2940  c0 4286  Ord word 6320  1oc1o 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-suc 6327  df-1o 8416
This theorem is referenced by:  1ellim  8448
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