Theorem nlim1 8526

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 8525	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5313	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4931	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3012	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8512	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4924	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3012	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2940	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1137	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 197	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6391	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 323	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1537   ≠ wne 2938  ∅c0 4339  {csn 4631  ∪ cuni 4912  Ord word 6385  Lim wlim 6387  1_oc1o 8498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-lim 6391  df-suc 6392  df-1o 8505

This theorem is referenced by:  1ellim  8535  2ellim  8536