Theorem onepsuc 44515

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 6472	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		onsuc 7858	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		epelg 5559	. . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴))
4	2, 3	syl 18	. 2 ⊢ (𝐴 ∈ On → (𝐴 E suc 𝐴 ↔ 𝐴 ∈ suc 𝐴))
5	1, 4	mpbird 260	1 ⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2147   class class class wbr 5108   E cep 5557  Oncon0 6388  suc csuc 6390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-suc 6394

This theorem is used by: (None)