Theorem nlim1 8497

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 8495	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5261	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4886	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3029	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8483	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4879	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3029	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2958	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1156	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 200	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6367	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 326	1 ⊢ ¬ Lim 1o

This proof depends on syntax axioms:  ¬ wn 3   ∧ w3a 1103   = wceq 1570   ≠ wne 2956  ∅c0 4279  {csn 4584  ∪ cuni 4867  Ord word 6361  Lim wlim 6363  1_oc1o 8469

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587  df-uni 4868  df-lim 6367  df-suc 6368  df-1o 8476

This theorem is used by:  1ellim  8506  2ellim  8507