Theorem 1oequni2o 37283

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 8519	. . 3 ⊢ 2o = suc 1o
2		2on 8532	. . . 4 ⊢ 2o ∈ On
3		2on0 8534	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8694	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7913	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 37282	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1461	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2764	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8530	. . 3 ⊢ 1o ∈ On
11		onuni 7820	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6501	. . 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 230	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2103   ≠ wne 2942  ∅c0 4347  ∪ cuni 4931  Oncon0 6394  Lim wlim 6395  suc csuc 6396  ωcom 7899  1_oc1o 8511  2_oc2o 8512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-tr 5287  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-om 7900  df-1o 8518  df-2o 8519

This theorem is referenced by:  finxpreclem4  37309