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Theorem 2ordpr 4276
Description: Version of 2on 6039 with the definition of 2𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Assertion
Ref Expression
2ordpr Ord {∅, {∅}}

Proof of Theorem 2ordpr
StepHypRef Expression
1 ord0 4155 . . 3 Ord ∅
2 ordsucim 4253 . . 3 (Ord ∅ → Ord suc ∅)
3 ordsucim 4253 . . 3 (Ord suc ∅ → Ord suc suc ∅)
41, 2, 3mp2b 8 . 2 Ord suc suc ∅
5 df-suc 4135 . . . 4 suc {∅} = ({∅} ∪ {{∅}})
6 suc0 4175 . . . . 5 suc ∅ = {∅}
7 suceq 4166 . . . . 5 (suc ∅ = {∅} → suc suc ∅ = suc {∅})
86, 7ax-mp 7 . . . 4 suc suc ∅ = suc {∅}
9 df-pr 3409 . . . 4 {∅, {∅}} = ({∅} ∪ {{∅}})
105, 8, 93eqtr4i 2086 . . 3 suc suc ∅ = {∅, {∅}}
11 ordeq 4136 . . 3 (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
1210, 11ax-mp 7 . 2 (Ord suc suc ∅ ↔ Ord {∅, {∅}})
134, 12mpbi 137 1 Ord {∅, {∅}}
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  cun 2942  c0 3251  {csn 3402  {cpr 3403  Ord word 4126  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-suc 4135
This theorem is referenced by:  ontr2exmid  4277  ordtri2or2exmidlem  4278  onsucelsucexmidlem  4281
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