Theorem 2ordpr 4267

Description: Version of 2on 6032 with the definition of 2_𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)

Assertion

Ref	Expression
2ordpr	⊢ Ord {∅, {∅}}

Proof of Theorem 2ordpr

Step	Hyp	Ref	Expression
1		ord0 4146	. . 3 ⊢ Ord ∅
2		ordsucim 4244	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4244	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4126	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4166	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4157	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 7	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3405	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2111	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4127	. . 3 ⊢ (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
12	10, 11	ax-mp 7	. 2 ⊢ (Ord suc suc ∅ ↔ Ord {∅, {∅}})
13	4, 12	mpbi 143	1 ⊢ Ord {∅, {∅}}

Syntax hints:   ↔ wb 103   = wceq 1284   ∪ cun 2971  ∅c0 3251  {csn 3398  {cpr 3399  Ord word 4117  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-suc 4126

This theorem is referenced by:  ontr2exmid  4268  ordtri2or2exmidlem  4269  onsucelsucexmidlem  4272