Theorem 1oequni2o 37701

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 8400	. . 3 ⊢ 2o = suc 1o
2		2on 8412	. . . 4 ⊢ 2o ∈ On
3		2on0 8413	. . . 4 ⊢ 2o ≠ ∅
4		2onn 8572	. . . . 5 ⊢ 2o ∈ ω
5		nnlim 7825	. . . . 5 ⊢ (2o ∈ ω → ¬ Lim 2o)
6	4, 5	ax-mp 5	. . . 4 ⊢ ¬ Lim 2o
7		onsucuni3 37700	. . . 4 ⊢ ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → 2o = suc ∪ 2o)
8	2, 3, 6, 7	mp3an 1464	. . 3 ⊢ 2o = suc ∪ 2o
9	1, 8	eqtr3i 2762	. 2 ⊢ suc 1o = suc ∪ 2o
10		1on 8411	. . 3 ⊢ 1o ∈ On
11		onuni 7736	. . . 4 ⊢ (2o ∈ On → ∪ 2o ∈ On)
12	2, 11	ax-mp 5	. . 3 ⊢ ∪ 2o ∈ On
13		suc11 6427	. . 3 ⊢ ((1o ∈ On ∧ ∪ 2o ∈ On) → (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o))
14	10, 12, 13	mp2an 693	. 2 ⊢ (suc 1o = suc ∪ 2o ↔ 1o = ∪ 2o)
15	9, 14	mpbi 230	1 ⊢ 1o = ∪ 2o

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  ∪ cuni 4851  Oncon0 6318  Lim wlim 6319  suc csuc 6320  ωcom 7811  1_oc1o 8392  2_oc2o 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7812  df-1o 8399  df-2o 8400

This theorem is referenced by:  finxpreclem4  37727