Theorem nlim1 8480

Description: 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)

Assertion

Ref	Expression
nlim1	⊢ ¬ Lim 1o

Proof of Theorem nlim1

Step	Hyp	Ref	Expression
1		1n0 8478	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5272	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4893	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3033	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8466	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4886	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3033	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2962	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1156	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 200	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6369	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 326	1 ⊢ ¬ Lim 1o

This proof depends on syntax axioms:  ¬ wn 3   ∧ w3a 1103   = wceq 1570   ≠ wne 2960  ∅c0 4286  {csn 4591  ∪ cuni 4874  Ord word 6363  Lim wlim 6365  1_oc1o 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-sn 4592  df-pr 4594  df-uni 4875  df-lim 6369  df-suc 6370  df-1o 8459

This theorem is used by:  1ellim  8489  2ellim  8490