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Theorem 2ordpr 4439
 Description: Version of 2on 6322 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 4313 . . 3 Ord ∅
2 ordsucim 4416 . . 3 (Ord ∅ → Ord suc ∅)
3 ordsucim 4416 . . 3 (Ord suc ∅ → Ord suc suc ∅)
41, 2, 3mp2b 8 . 2 Ord suc suc ∅
5 df-suc 4293 . . . 4 suc {∅} = ({∅} ∪ {{∅}})
6 suc0 4333 . . . . 5 suc ∅ = {∅}
7 suceq 4324 . . . . 5 (suc ∅ = {∅} → suc suc ∅ = suc {∅})
86, 7ax-mp 5 . . . 4 suc suc ∅ = suc {∅}
9 df-pr 3534 . . . 4 {∅, {∅}} = ({∅} ∪ {{∅}})
105, 8, 93eqtr4i 2170 . . 3 suc suc ∅ = {∅, {∅}}
11 ordeq 4294 . . 3 (suc suc ∅ = {∅, {∅}} → (Ord suc suc ∅ ↔ Ord {∅, {∅}}))
1210, 11ax-mp 5 . 2 (Ord suc suc ∅ ↔ Ord {∅, {∅}})
134, 12mpbi 144 1 Ord {∅, {∅}}
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331   ∪ cun 3069  ∅c0 3363  {csn 3527  {cpr 3528  Ord word 4284  suc csuc 4287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-tr 4027  df-iord 4288  df-suc 4293 This theorem is referenced by:  ontr2exmid  4440  ordtri2or2exmidlem  4441  onsucelsucexmidlem  4444
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