Theorem nlim1 8511

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 8509	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5264	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4889	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3032	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8497	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4882	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3032	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2961	. . 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 6374	. 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 1569   ≠ wne 2959  ∅c0 4281  {csn 4588  ∪ cuni 4870  Ord word 6368  Lim wlim 6370  1_oc1o 8483

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4871  df-lim 6374  df-suc 6375  df-1o 8490

This theorem is used by:  1ellim  8520  2ellim  8521  cflim2  10354