Theorem onepsuc 44093

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 6441	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		onsuc 7809	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		epelg 5556	. . 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 2145   class class class wbr 5103   E cep 5554  Oncon0 6357  suc csuc 6359

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7736

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-suc 6363

This theorem is used by: (None)