Theorem nlim1 8545

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 8544	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5325	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4950	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3020	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8529	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4943	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3020	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2948	. . 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 6400	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1537   ≠ wne 2946  ∅c0 4352  {csn 4648  ∪ cuni 4931  Ord word 6394  Lim wlim 6396  1_oc1o 8515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-uni 4932  df-lim 6400  df-suc 6401  df-1o 8522

This theorem is referenced by:  1ellim  8554  2ellim  8555