Theorem 1oequni2o 35151

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 8125	. . 3 ⊢ 2o = suc 1o
2		2on 8132	. . . 4 ⊢ 2o ∈ On
3		2on0 8133	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8290	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7606	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 35150	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1462	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2763	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8131	. . 3 ⊢ 1o ∈ On
11		onuni 7521	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6269	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 692	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 233	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  ∅c0 4209  ∪ cuni 4793  Oncon0 6166  Lim wlim 6167  suc csuc 6168  ωcom 7593  1_oc1o 8117  2_oc2o 8118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-tr 5134  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-om 7594  df-1o 8124  df-2o 8125

This theorem is referenced by:  finxpreclem4  35177