Theorem epsoon 43607

Description: The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7730 and weso 5623. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epsoon	⊢ E Or On

Proof of Theorem epsoon

Step	Hyp	Ref	Expression
1		epweon 7730	. 2 ⊢ E We On
2		weso 5623	. 2 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	1 ⊢ E Or On

Syntax hints:   E cep 5531   Or wor 5539   We wwe 5584  Oncon0 6325

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  oneptri  43611