Theorem epsoon 44349

Description: The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7790 and weso 5641. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epsoon	⊢ E Or On

Proof of Theorem epsoon

Step	Hyp	Ref	Expression
1		epweon 7790	. 2 ⊢ E We On
2		weso 5641	. 2 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	1 ⊢ E Or On

This proof depends on syntax axioms:   E cep 5549   Or wor 5557   We wwe 5602  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by:  oneptri  44353