Theorem 2ordpr 4572

Description: Version of 2on 6511 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 4438	. . 3 ⊢ Ord ∅
2		ordsucim 4548	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4548	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4418	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4458	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4449	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 5	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3640	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2236	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4419	. . 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 1373   ∪ cun 3164  ∅c0 3460  {csn 3633  {cpr 3634  Ord word 4409  suc csuc 4412

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-suc 4418

This theorem is referenced by:  ontr2exmid  4573  ordtri2or2exmidlem  4574  onsucelsucexmidlem  4577