Theorem epsoon 43286

Description: The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7703 and weso 5602. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epsoon	⊢ E Or On

Proof of Theorem epsoon

Step	Hyp	Ref	Expression
1		epweon 7703	. 2 ⊢ E We On
2		weso 5602	. 2 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	1 ⊢ E Or On

Syntax hints:   E cep 5510   Or wor 5518   We wwe 5563  Oncon0 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305

This theorem is referenced by:  oneptri  43290