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Theorem onelssex 33191
 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 3916 . . 3 𝐴𝐴
2 sseq2 3920 . . . 4 (𝑏 = 𝐴 → (𝐴𝑏𝐴𝐴))
32rspcev 3543 . . 3 ((𝐴𝐶𝐴𝐴) → ∃𝑏𝐶 𝐴𝑏)
41, 3mpan2 690 . 2 (𝐴𝐶 → ∃𝑏𝐶 𝐴𝑏)
5 ontr2 6221 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑏𝑏𝐶) → 𝐴𝐶))
65expcomd 420 . . 3 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏𝐶 → (𝐴𝑏𝐴𝐶)))
76rexlimdv 3207 . 2 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏𝐶 𝐴𝑏𝐴𝐶))
84, 7impbid2 229 1 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐶 ↔ ∃𝑏𝐶 𝐴𝑏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3860  Oncon0 6174 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177  df-on 6178 This theorem is referenced by:  madebdayim  33661  madebdaylemold  33669
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