Theorem onexgt 43967

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 7805	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		sucidg 6444	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3		eleq2 2852	. . 3 ⊢ (𝑥 = suc 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴))
4	3	rspcev 3581	. 2 ⊢ ((suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
5	1, 2, 4	syl2anc 595	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ∈ wcel 2143  ∃wrex 3089  Oncon0 6360  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-suc 6366

This theorem is referenced by: (None)