Theorem onexgt 43778

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 7788	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		sucidg 6424	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3		eleq2 2850	. . 3 ⊢ (𝑥 = suc 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴))
4	3	rspcev 3580	. 2 ⊢ ((suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
5	1, 2, 4	syl2anc 593	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ∈ wcel 2141  ∃wrex 3085  Oncon0 6341  suc csuc 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345  df-suc 6347

This theorem is referenced by: (None)