Theorem nlim1 8414

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 8413	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5250	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4880	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3003	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8402	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4873	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3003	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2932	. . 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 6320	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1541   ≠ wne 2930  ∅c0 4283  {csn 4578  ∪ cuni 4861  Ord word 6314  Lim wlim 6316  1_oc1o 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-uni 4862  df-lim 6320  df-suc 6321  df-1o 8395

This theorem is referenced by:  1ellim  8423  2ellim  8424