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Theorem onepsuc 43242
Description: Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
Assertion
Ref Expression
onepsuc (𝐴 ∈ On → 𝐴 E suc 𝐴)

Proof of Theorem onepsuc
StepHypRef Expression
1 sucidg 6463 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 onsuc 7827 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
3 epelg 5583 . . 3 (suc 𝐴 ∈ On → (𝐴 E suc 𝐴𝐴 ∈ suc 𝐴))
42, 3syl 17 . 2 (𝐴 ∈ On → (𝐴 E suc 𝐴𝐴 ∈ suc 𝐴))
51, 4mpbird 257 1 (𝐴 ∈ On → 𝐴 E suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108   class class class wbr 5141   E cep 5581  Oncon0 6382  suc csuc 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-tr 5258  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-ord 6385  df-on 6386  df-suc 6388
This theorem is referenced by: (None)
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