Theorem epirron 43843

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 7762	. . 3 ⊢ E We On
2		weso 5643	. . 3 ⊢ ( E We On → E Or On)
3		sopo 5579	. . 3 ⊢ ( E Or On → E Po On)
4	1, 2, 3	mp2b 10	. 2 ⊢ E Po On
5		poirr 5572	. 2 ⊢ (( E Po On ∧ 𝐴 ∈ On) → ¬ 𝐴 E 𝐴)
6	4, 5	mpan 702	1 ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2145   class class class wbr 5105   E cep 5551   Po wpo 5558   Or wor 5559   We wwe 5604  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354

This theorem is referenced by: (None)