Theorem epirron 43700

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 7722	. . 3 ⊢ E We On
2		weso 5615	. . 3 ⊢ ( E We On → E Or On)
3		sopo 5551	. . 3 ⊢ ( E Or On → E Po On)
4	1, 2, 3	mp2b 10	. 2 ⊢ E Po On
5		poirr 5544	. 2 ⊢ (( E Po On ∧ 𝐴 ∈ On) → ¬ 𝐴 E 𝐴)
6	4, 5	mpan 691	1 ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   class class class wbr 5086   E cep 5523   Po wpo 5530   Or wor 5531   We wwe 5576  Oncon0 6317

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 5231  ax-pr 5370

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321

This theorem is referenced by: (None)