Theorem nlim1 8426

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 8425	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5254	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4884	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3006	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8414	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4877	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3006	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2935	. . 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 6330	. 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 2933  ∅c0 4287  {csn 4582  ∪ cuni 4865  Ord word 6324  Lim wlim 6326  1_oc1o 8400

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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-sn 4583  df-pr 4585  df-uni 4866  df-lim 6330  df-suc 6331  df-1o 8407

This theorem is referenced by:  1ellim  8435  2ellim  8436