Theorem nlim1 8528

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 8527	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5306	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4925	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3013	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8514	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4918	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3013	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2941	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1138	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 197	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6388	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1539   ≠ wne 2939  ∅c0 4332  {csn 4625  ∪ cuni 4906  Ord word 6382  Lim wlim 6384  1_oc1o 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-lim 6388  df-suc 6389  df-1o 8507

This theorem is referenced by:  1ellim  8537  2ellim  8538