Theorem onelssex 6402

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 3996	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq2 4000	. . . 4 ⊢ (𝑏 = 𝐴 → (𝐴 ⊆ 𝑏 ↔ 𝐴 ⊆ 𝐴))
3	2	rspcev 3604	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐴) → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
4	1, 3	mpan2 688	. 2 ⊢ (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
5		ontr2 6401	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝑏 ∧ 𝑏 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	expcomd 416	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏 ∈ 𝐶 → (𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶)))
7	6	rexlimdv 3145	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶))
8	4, 7	impbid2 225	1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  ∃wrex 3062   ⊆ wss 3940  Oncon0 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358

This theorem is referenced by:  madebdayim  27730  madebdaylemold  27740