Theorem 2ordpr 4249

Description: Version of 2on 6009 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 4128	. . 3 ⊢ Ord ∅
2		ordsucim 4226	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4226	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4108	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4148	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4139	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 7	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3382	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2070	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4109	. . 3 ⊢ (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
12	10, 11	ax-mp 7	. 2 ⊢ (Ord suc suc ∅ ↔ Ord {∅, {∅}})
13	4, 12	mpbi 133	1 ⊢ Ord {∅, {∅}}

Syntax hints:   ↔ wb 98   = wceq 1243   ∪ cun 2915  ∅c0 3224  {csn 3375  {cpr 3376  Ord word 4099  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108

This theorem is referenced by:  ontr2exmid  4250  ordtri2or2exmidlem  4251  onsucelsucexmidlem  4254