Theorem 1oequni2o 36880

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 8494	. . 3 ⊢ 2o = suc 1o
2		2on 8507	. . . 4 ⊢ 2o ∈ On
3		2on0 8509	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8669	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7890	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 36879	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1457	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2758	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8505	. . 3 ⊢ 1o ∈ On
11		onuni 7797	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6481	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 690	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 229	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∅c0 4326  ∪ cuni 4912  Oncon0 6374  Lim wlim 6375  suc csuc 6376  ωcom 7876  1_oc1o 8486  2_oc2o 8487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-om 7877  df-1o 8493  df-2o 8494

This theorem is referenced by:  finxpreclem4  36906