Theorem epirron 42468

Description: The strict order on the ordinals is irreflexive. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
epirron	⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Proof of Theorem epirron

Step	Hyp	Ref	Expression
1		epweon 7766	. . 3 ⊢ E We On
2		weso 5667	. . 3 ⊢ ( E We On → E Or On)
3		sopo 5607	. . 3 ⊢ ( E Or On → E Po On)
4	1, 2, 3	mp2b 10	. 2 ⊢ E Po On
5		poirr 5600	. 2 ⊢ (( E Po On ∧ 𝐴 ∈ On) → ¬ 𝐴 E 𝐴)
6	4, 5	mpan 687	1 ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2105   class class class wbr 5148   E cep 5579   Po wpo 5586   Or wor 5587   We wwe 5630  Oncon0 6364

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368

This theorem is referenced by: (None)