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Theorem ordgt0ge1 6455
Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
ordgt0ge1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))

Proof of Theorem ordgt0ge1
StepHypRef Expression
1 0elon 4407 . . 3 ∅ ∈ On
2 ordelsuc 4519 . . 3 ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
31, 2mpan 424 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴))
4 df-1o 6436 . . 3 1o = suc ∅
54sseq1i 3196 . 2 (1o𝐴 ↔ suc ∅ ⊆ 𝐴)
63, 5bitr4di 198 1 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2160  wss 3144  c0 3437  Ord word 4377  Oncon0 4378  suc csuc 4380  1oc1o 6429
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-1o 6436
This theorem is referenced by:  ordge1n0im  6456  archnqq  7441
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