Theorem epirron 43708

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 7719	. . 3 ⊢ E We On
2		weso 5610	. . 3 ⊢ ( E We On → E Or On)
3		sopo 5546	. . 3 ⊢ ( E Or On → E Po On)
4	1, 2, 3	mp2b 10	. 2 ⊢ E Po On
5		poirr 5539	. 2 ⊢ (( E Po On ∧ 𝐴 ∈ On) → ¬ 𝐴 E 𝐴)
6	4, 5	mpan 696	1 ⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2119   class class class wbr 5073   E cep 5518   Po wpo 5525   Or wor 5526   We wwe 5571  Oncon0 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-tr 5181  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6314  df-on 6315

This theorem is referenced by: (None)