Theorem nlim1 8491

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 8490	. . . . . 6 ⊢ 1o ≠ ∅
2		0ex 5307	. . . . . . 7 ⊢ ∅ ∈ V
3	2	unisn 4930	. . . . . 6 ⊢ ∪ {∅} = ∅
4	1, 3	neeqtrri 3014	. . . . 5 ⊢ 1o ≠ ∪ {∅}
5		df1o2 8475	. . . . . 6 ⊢ 1o = {∅}
6	5	unieqi 4921	. . . . 5 ⊢ ∪ 1o = ∪ {∅}
7	4, 6	neeqtrri 3014	. . . 4 ⊢ 1o ≠ ∪ 1o
8	7	neii 2942	. . 3 ⊢ ¬ 1o = ∪ 1o
9		simp3 1138	. . 3 ⊢ ((Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o) → 1o = ∪ 1o)
10	8, 9	mto 196	. 2 ⊢ ¬ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o)
11		df-lim 6369	. 2 ⊢ (Lim 1o ↔ (Ord 1o ∧ 1o ≠ ∅ ∧ 1o = ∪ 1o))
12	10, 11	mtbir 322	1 ⊢ ¬ Lim 1o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1541   ≠ wne 2940  ∅c0 4322  {csn 4628  ∪ cuni 4908  Ord word 6363  Lim wlim 6365  1_oc1o 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-lim 6369  df-suc 6370  df-1o 8468

This theorem is referenced by:  1ellim  8500  2ellim  8501