Theorem 2ordpr 4561

Description: Version of 2on 6492 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 4427	. . 3 ⊢ Ord ∅
2		ordsucim 4537	. . 3 ⊢ (Ord ∅ → Ord suc ∅)
3		ordsucim 4537	. . 3 ⊢ (Ord suc ∅ → Ord suc suc ∅)
4	1, 2, 3	mp2b 8	. 2 ⊢ Ord suc suc ∅
5		df-suc 4407	. . . 4 ⊢ suc {∅} = ({∅} ∪ {{∅}})
6		suc0 4447	. . . . 5 ⊢ suc ∅ = {∅}
7		suceq 4438	. . . . 5 ⊢ (suc ∅ = {∅} → suc suc ∅ = suc {∅})
8	6, 7	ax-mp 5	. . . 4 ⊢ suc suc ∅ = suc {∅}
9		df-pr 3630	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
10	5, 8, 9	3eqtr4i 2227	. . 3 ⊢ suc suc ∅ = {∅, {∅}}
11		ordeq 4408	. . 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 1364   ∪ cun 3155  ∅c0 3451  {csn 3623  {cpr 3624  Ord word 4398  suc csuc 4401

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-suc 4407

This theorem is referenced by:  ontr2exmid  4562  ordtri2or2exmidlem  4563  onsucelsucexmidlem  4566