Theorem 1oequni2o 38073

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 8460	. . 3 ⊢ 2o = suc 1o
2		2on 8473	. . . 4 ⊢ 2o ∈ On
3		2on0 8474	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8634	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7882	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 38072	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1490	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2790	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8472	. . 3 ⊢ 1o ∈ On
11		onuni 7793	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6474	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 705	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 233	1 ⊢ 1o = ∪ 2o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∅c0 4286  ∪ cuni 4874  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ωcom 7868  1_oc1o 8452  2_oc2o 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7869  df-1o 8459  df-2o 8460

This theorem is used by:  finxpreclem4  38099