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

Step	Hyp	Ref	Expression
1		sucidg 6445	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		onsuc 7813	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		epelg 5565	. . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2107   class class class wbr 5123   E cep 5563  Oncon0 6363  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-tr 5240  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367  df-suc 6369

This theorem is referenced by: (None)