Theorem epsoon 43865

Description: The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7770 and weso 5650. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epsoon	⊢ E Or On

Proof of Theorem epsoon

Step	Hyp	Ref	Expression
1		epweon 7770	. 2 ⊢ E We On
2		weso 5650	. 2 ⊢ ( E We On → E Or On)
3	1, 2	ax-mp 5	1 ⊢ E Or On

Syntax hints:   E cep 5558   Or wor 5566   We wwe 5611  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6360  df-on 6361

This theorem is referenced by:  oneptri  43869