Theorem 1oequni2o 33714

Description: The ordinal number 1_𝑜 is the predecessor of the ordinal number 2_𝑜. (Contributed by ML, 19-Oct-2020.)

Assertion

Ref	Expression
1oequni2o	⊢ 1𝑜 = ∪ 2𝑜

Proof of Theorem 1oequni2o

Step	Hyp	Ref	Expression
1		df-2o 7800	. . 3 ⊢ 2𝑜 = suc 1𝑜
2		2on 7808	. . . 4 ⊢ 2𝑜 ∈ On
3		2on0 7809	. . . 4 ⊢ 2𝑜 ≠ ∅
4		2onn 7960	. . . . 5 ⊢ 2𝑜 ∈ ω
5		nnlim 7312	. . . . 5 ⊢ (2𝑜 ∈ ω → ¬ Lim 2𝑜)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2𝑜
7		onsucuni3 33713	. . . 4 ⊢ ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → 2𝑜 = suc ∪ 2𝑜)
8	2, 3, 6, 7	mp3an 1586	. . 3 ⊢ 2𝑜 = suc ∪ 2𝑜
9	1, 8	eqtr3i 2823	. 2 ⊢ suc 1𝑜 = suc ∪ 2𝑜
10		1on 7806	. . 3 ⊢ 1𝑜 ∈ On
11		onuni 7227	. . . 4 ⊢ (2𝑜 ∈ On → ∪ 2𝑜 ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2𝑜 ∈ On
13		suc11 6044	. . 3 ⊢ ((1𝑜 ∈ On ∧ ∪ 2𝑜 ∈ On) → (suc 1𝑜 = suc ∪ 2𝑜 ↔ 1𝑜 = ∪ 2𝑜))
14	10, 12, 13	mp2an 684	. 2 ⊢ (suc 1𝑜 = suc ∪ 2𝑜 ↔ 1𝑜 = ∪ 2𝑜)
15	9, 14	mpbi 222	1 ⊢ 1𝑜 = ∪ 2𝑜

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1653   ∈ wcel 2157   ≠ wne 2971  ∅c0 4115  ∪ cuni 4628  Oncon0 5941  Lim wlim 5942  suc csuc 5943  ωcom 7299  1_𝑜c1o 7792  2_𝑜c2o 7793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-om 7300  df-1o 7799  df-2o 7800

This theorem is referenced by:  finxpreclem4  33729