Theorem epirron 43243

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 7794	. . 3 ⊢ E We On
2		weso 5680	. . 3 ⊢ ( E We On → E Or On)
3		sopo 5616	. . 3 ⊢ ( E Or On → E Po On)
4	1, 2, 3	mp2b 10	. 2 ⊢ E Po On
5		poirr 5609	. 2 ⊢ (( E Po On ∧ 𝐴 ∈ On) → ¬ 𝐴 E 𝐴)
6	4, 5	mpan 690	1 ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   class class class wbr 5148   E cep 5588   Po wpo 5595   Or wor 5596   We wwe 5640  Oncon0 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390

This theorem is referenced by: (None)