Theorem onepsuc 41934

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 6442	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		onsuc 7794	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		epelg 5580	. . 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 205   ∈ wcel 2107   class class class wbr 5147   E cep 5578  Oncon0 6361  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367

This theorem is referenced by: (None)