Theorem onexgt 43586

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 7765	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		sucidg 6408	. 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3		eleq2 2826	. . 3 ⊢ (𝑥 = suc 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴))
4	3	rspcev 3578	. 2 ⊢ ((suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
5	1, 2, 4	syl2anc 585	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ∈ wcel 2114  ∃wrex 3062  Oncon0 6325  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-suc 6331

This theorem is referenced by: (None)