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Theorem onelssex 6432
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) → (𝐴𝐶 ↔ ∃𝑏𝐶 𝐴𝑏))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏

Proof of Theorem onelssex
StepHypRef Expression
1 ssid 4006 . . 3 𝐴𝐴
2 sseq2 4010 . . . 4 (𝑏 = 𝐴 → (𝐴𝑏𝐴𝐴))
32rspcev 3622 . . 3 ((𝐴𝐶𝐴𝐴) → ∃𝑏𝐶 𝐴𝑏)
41, 3mpan2 691 . 2 (𝐴𝐶 → ∃𝑏𝐶 𝐴𝑏)
5 ontr2 6431 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑏𝑏𝐶) → 𝐴𝐶))
65expcomd 416 . . 3 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏𝐶 → (𝐴𝑏𝐴𝐶)))
76rexlimdv 3153 . 2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏𝐶 𝐴𝑏𝐴𝐶))
84, 7impbid2 226 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐶 ↔ ∃𝑏𝐶 𝐴𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wrex 3070  wss 3951  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  madebdayim  27926  madebdaylemold  27936
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