Theorem nlim1 8407

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 8406	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5246	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4877	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 2998	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8395	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4870	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 2998	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2927	. . 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 6312	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1540   ≠ wne 2925  ∅c0 4284  {csn 4577  ∪ cuni 4858  Ord word 6306  Lim wlim 6308  1_oc1o 8381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-sn 4578  df-pr 4580  df-uni 4859  df-lim 6312  df-suc 6313  df-1o 8388

This theorem is referenced by:  1ellim  8416  2ellim  8417