Theorem nlim1 8458

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 8456	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5257	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4884	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3030	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8444	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4877	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3030	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2959	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1151	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 199	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6351	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 325	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1098   = wceq 1560   ≠ wne 2957  ∅c0 4285  {csn 4582  ∪ cuni 4865  Ord word 6345  Lim wlim 6347  1_oc1o 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4583  df-pr 4585  df-uni 4866  df-lim 6351  df-suc 6352  df-1o 8437

This theorem is referenced by:  1ellim  8467  2ellim  8468