Theorem 1oequni2o 37356

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 8435	. . 3 ⊢ 2o = suc 1o
2		2on 8447	. . . 4 ⊢ 2o ∈ On
3		2on0 8448	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8606	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7856	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 37355	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1463	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2754	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8446	. . 3 ⊢ 1o ∈ On
11		onuni 7764	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6441	. . 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 230	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  ∪ cuni 4871  Oncon0 6332  Lim wlim 6333  suc csuc 6334  ωcom 7842  1_oc1o 8427  2_oc2o 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-om 7843  df-1o 8434  df-2o 8435

This theorem is referenced by:  finxpreclem4  37382