Theorem 2ordpr 4645

Description: Version of 2on 6655 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 4511	. . 3 ⊢ Ord ∅
2		ordsucim 4621	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4621	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4491	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4531	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4522	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 5	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3695	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2263	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4492	. . 3 ⊢ (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
12	10, 11	ax-mp 5	. 2 ⊢ (Ord suc suc ∅ ↔ Ord {∅, {∅}})
13	4, 12	mpbi 145	1 ⊢ Ord {∅, {∅}}

Syntax hints:   ↔ wb 105   = wceq 1398   ∪ cun 3208  ∅c0 3507  {csn 3688  {cpr 3689  Ord word 4482  suc csuc 4485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-uni 3914  df-tr 4208  df-iord 4486  df-suc 4491

This theorem is referenced by:  ontr2exmid  4646  ordtri2or2exmidlem  4647  onsucelsucexmidlem  4650