Theorem nlim1 8474

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 8472	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5272	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4895	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3037	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8460	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4888	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3037	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2966	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1154	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 200	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6366	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 326	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1101   = wceq 1567   ≠ wne 2964  ∅c0 4294  {csn 4594  ∪ cuni 4876  Ord word 6360  Lim wlim 6362  1_oc1o 8446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-uni 4877  df-lim 6366  df-suc 6367  df-1o 8453

This theorem is referenced by:  1ellim  8483  2ellim  8484