Theorem 1oequni2o 33216

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 7561	. . 3 ⊢ 2𝑜 = suc 1𝑜
2		2on 7568	. . . 4 ⊢ 2𝑜 ∈ On
3		2on0 7569	. . . 4 ⊢ 2𝑜 ≠ ∅
4		2onn 7720	. . . . 5 ⊢ 2𝑜 ∈ ω
5		nnlim 7078	. . . . 5 ⊢ (2𝑜 ∈ ω → ¬ Lim 2𝑜)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2𝑜
7		onsucuni3 33215	. . . 4 ⊢ ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → 2𝑜 = suc ∪ 2𝑜)
8	2, 3, 6, 7	mp3an 1424	. . 3 ⊢ 2𝑜 = suc ∪ 2𝑜
9	1, 8	eqtr3i 2646	. 2 ⊢ suc 1𝑜 = suc ∪ 2𝑜
10		suc11reg 8516	. 2 ⊢ (suc 1𝑜 = suc ∪ 2𝑜 ↔ 1𝑜 = ∪ 2𝑜)
11	9, 10	mpbi 220	1 ⊢ 1𝑜 = ∪ 2𝑜

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ∪ cuni 4436  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-1o 7560  df-2o 7561

This theorem is referenced by:  finxpreclem4  33231