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 5229	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4857	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3007	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8402	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4850	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3007	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2936	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1144	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 198	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6315	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 324	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1092   = wceq 1547   ≠ wne 2934  ∅c0 4261  {csn 4555  ∪ cuni 4838  Ord word 6309  Lim wlim 6311  1_oc1o 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-sn 4556  df-pr 4558  df-uni 4839  df-lim 6315  df-suc 6316  df-1o 8395

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