Theorem 1oequni2o 36705

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 8462	. . 3 ⊢ 2o = suc 1o
2		2on 8475	. . . 4 ⊢ 2o ∈ On
3		2on0 8477	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8636	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7862	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 36704	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1457	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2754	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8473	. . 3 ⊢ 1o ∈ On
11		onuni 7769	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6461	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 689	. 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 2932  ∅c0 4314  ∪ cuni 4899  Oncon0 6354  Lim wlim 6355  suc csuc 6356  ωcom 7848  1_oc1o 8454  2_oc2o 8455

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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-om 7849  df-1o 8461  df-2o 8462

This theorem is referenced by:  finxpreclem4  36731