Theorem 1oequni2o 36238

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 8464	. . 3 ⊢ 2o = suc 1o
2		2on 8477	. . . 4 ⊢ 2o ∈ On
3		2on0 8479	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8638	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7866	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 36237	. . . 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 8475	. . 3 ⊢ 1o ∈ On
11		onuni 7773	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6469	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 691	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 229	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∅c0 4322  ∪ cuni 4908  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ωcom 7852  1_oc1o 8456  2_oc2o 8457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-om 7853  df-1o 8463  df-2o 8464

This theorem is referenced by:  finxpreclem4  36264