Theorem onepsuc 43704

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 6400	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		onsuc 7760	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		epelg 5526	. . 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 258	1 ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2119   class class class wbr 5079   E cep 5524  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by: (None)