Theorem nlim1 8470

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 8468	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5270	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4891	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3031	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8456	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4884	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3031	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2960	. . 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 6365	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 326	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1103   = wceq 1570   ≠ wne 2958  ∅c0 4286  {csn 4589  ∪ cuni 4872  Ord word 6359  Lim wlim 6361  1_oc1o 8442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4590  df-pr 4592  df-uni 4873  df-lim 6365  df-suc 6366  df-1o 8449

This theorem is referenced by:  1ellim  8479  2ellim  8480