Theorem epsoon 43681

Description: The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7729 and weso 5622. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epsoon	⊢ E Or On

Proof of Theorem epsoon

Step	Hyp	Ref	Expression
1		epweon 7729	. 2 ⊢ E We On
2		weso 5622	. 2 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	1 ⊢ E Or On

Syntax hints:   E cep 5530   Or wor 5538   We wwe 5583  Oncon0 6323

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  oneptri  43685