Theorem 1oequni2o 37862

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 8438	. . 3 ⊢ 2o = suc 1o
2		2on 8451	. . . 4 ⊢ 2o ∈ On
3		2on0 8452	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8612	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7860	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 37861	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1482	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2787	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8450	. . 3 ⊢ 1o ∈ On
11		onuni 7771	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6455	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 702	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 232	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∅c0 4285  ∪ cuni 4865  Oncon0 6346  Lim wlim 6347  suc csuc 6348  ωcom 7846  1_oc1o 8430  2_oc2o 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-om 7847  df-1o 8437  df-2o 8438

This theorem is referenced by:  finxpreclem4  37888