Theorem nlim1 8424

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 8423	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5242	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4869	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3005	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8412	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4862	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3005	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2934	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1139	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 197	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6328	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1542   ≠ wne 2932  ∅c0 4273  {csn 4567  ∪ cuni 4850  Ord word 6322  Lim wlim 6324  1_oc1o 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-sn 4568  df-pr 4570  df-uni 4851  df-lim 6328  df-suc 6329  df-1o 8405

This theorem is referenced by:  1ellim  8433  2ellim  8434