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Theorem oneptri 43247
Description: The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
Assertion
Ref Expression
oneptri ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐵 E 𝐴𝐴 = 𝐵))

Proof of Theorem oneptri
StepHypRef Expression
1 epsoon 43243 . . 3 E Or On
2 sotrieq 5621 . . 3 (( E Or On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵𝐵 E 𝐴)))
31, 2mpan 690 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵𝐵 E 𝐴)))
4 xoror 1518 . . 3 (((𝐴 E 𝐵𝐵 E 𝐴) ⊻ 𝐴 = 𝐵) → ((𝐴 E 𝐵𝐵 E 𝐴) ∨ 𝐴 = 𝐵))
5 xorcom 1514 . . . 4 (((𝐴 E 𝐵𝐵 E 𝐴) ⊻ 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ⊻ (𝐴 E 𝐵𝐵 E 𝐴)))
6 df-xor 1512 . . . 4 ((𝐴 = 𝐵 ⊻ (𝐴 E 𝐵𝐵 E 𝐴)) ↔ ¬ (𝐴 = 𝐵 ↔ (𝐴 E 𝐵𝐵 E 𝐴)))
7 xor3 382 . . . 4 (¬ (𝐴 = 𝐵 ↔ (𝐴 E 𝐵𝐵 E 𝐴)) ↔ (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵𝐵 E 𝐴)))
85, 6, 73bitrri 298 . . 3 ((𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵𝐵 E 𝐴)) ↔ ((𝐴 E 𝐵𝐵 E 𝐴) ⊻ 𝐴 = 𝐵))
9 df-3or 1088 . . 3 ((𝐴 E 𝐵𝐵 E 𝐴𝐴 = 𝐵) ↔ ((𝐴 E 𝐵𝐵 E 𝐴) ∨ 𝐴 = 𝐵))
104, 8, 93imtr4i 292 . 2 ((𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵𝐵 E 𝐴)) → (𝐴 E 𝐵𝐵 E 𝐴𝐴 = 𝐵))
113, 10syl 17 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐵 E 𝐴𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3o 1086  wxo 1511   = wceq 1540  wcel 2108   class class class wbr 5141   E cep 5581   Or wor 5589  Oncon0 6382
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  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 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-tr 5258  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-ord 6385  df-on 6386
This theorem is referenced by: (None)
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