Theorem onexgt 43252

Description: For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexgt

Step	Hyp	Ref	Expression
1		onsuc 7831	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		sucidg 6465	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3		eleq2 2830	. . 3 ⊢ (𝑥 = suc 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴))
4	3	rspcev 3622	. 2 ⊢ ((suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
5	1, 2, 4	syl2anc 584	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃wrex 3070  Oncon0 6384  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by: (None)