Theorem onelssex 6405

Description: Ordinal less than is equivalent to having an ordinal between them. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
onelssex	⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏

Proof of Theorem onelssex

Step	Hyp	Ref	Expression
1		ssid 3953	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq2 3957	. . . 4 ⊢ (𝑏 = 𝐴 → (𝐴 ⊆ 𝑏 ↔ 𝐴 ⊆ 𝐴))
3	2	rspcev 3577	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐴) → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
4	1, 3	mpan2 704	. 2 ⊢ (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
5		ontr2 6404	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝑏 ∧ 𝑏 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	expcomd 422	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏 ∈ 𝐶 → (𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶)))
7	6	rexlimdv 3162	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶))
8	4, 7	impbid2 229	1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∃wrex 3087   ⊆ wss 3899  Oncon0 6355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6358  df-on 6359

This theorem is used by:  madebdayim  28256  madebdaylemold  28266