Theorem 2ordpr 4368

Description: Version of 2on 6228 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 4242	. . 3 ⊢ Ord ∅
2		ordsucim 4345	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4345	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4222	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4262	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4253	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 7	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3473	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2125	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4223	. . 3 ⊢ (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
12	10, 11	ax-mp 7	. 2 ⊢ (Ord suc suc ∅ ↔ Ord {∅, {∅}})
13	4, 12	mpbi 144	1 ⊢ Ord {∅, {∅}}

Syntax hints:   ↔ wb 104   = wceq 1296   ∪ cun 3011  ∅c0 3302  {csn 3466  {cpr 3467  Ord word 4213  suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-suc 4222

This theorem is referenced by:  ontr2exmid  4369  ordtri2or2exmidlem  4370  onsucelsucexmidlem  4373