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Theorem oneptr 43272
Description: The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
Assertion
Ref Expression
oneptr ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵𝐵 E 𝐶) → 𝐴 E 𝐶))

Proof of Theorem oneptr
StepHypRef Expression
1 epweon 7796 . 2 E We On
2 weso 5675 . 2 ( E We On → E Or On)
3 sopo 5610 . . 3 ( E Or On → E Po On)
4 potr 5604 . . . 4 (( E Po On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 E 𝐵𝐵 E 𝐶) → 𝐴 E 𝐶))
54ex 412 . . 3 ( E Po On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵𝐵 E 𝐶) → 𝐴 E 𝐶)))
63, 5syl 17 . 2 ( E Or On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵𝐵 E 𝐶) → 𝐴 E 𝐶)))
71, 2, 6mp2b 10 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵𝐵 E 𝐶) → 𝐴 E 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2107   class class class wbr 5142   E cep 5582   Po wpo 5589   Or wor 5590   We wwe 5635  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by: (None)
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