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Theorem onelssex 6434
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 4018 . . 3 𝐴𝐴
2 sseq2 4022 . . . 4 (𝑏 = 𝐴 → (𝐴𝑏𝐴𝐴))
32rspcev 3622 . . 3 ((𝐴𝐶𝐴𝐴) → ∃𝑏𝐶 𝐴𝑏)
41, 3mpan2 691 . 2 (𝐴𝐶 → ∃𝑏𝐶 𝐴𝑏)
5 ontr2 6433 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑏𝑏𝐶) → 𝐴𝐶))
65expcomd 416 . . 3 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏𝐶 → (𝐴𝑏𝐴𝐶)))
76rexlimdv 3151 . 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 2106  wrex 3068  wss 3963  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  madebdayim  27941  madebdaylemold  27951
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