Theorem onprcALT 36532

Description: Alternate proof of onprc 7792 from onprcr 36531. (Contributed by Adrian Ducourtial, 15-Oct-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onprcALT	⊢ ¬ On ∈ V

Proof of Theorem onprcALT

Step	Hyp	Ref	Expression
1		df-on 6366	. . 3 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		rabab 3480	. . 3 ⊢ {𝑥 ∈ V ∣ Ord 𝑥} = {𝑥 ∣ Ord 𝑥}
3	1, 2	eqtr4i 2786	. 2 ⊢ On = {𝑥 ∈ V ∣ Ord 𝑥}
4		tron 6385	. 2 ⊢ Tr On
5	3, 4	onprcr 36531	1 ⊢ ¬ On ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2144  {cab 2738  {crab 3412  Vcvv 3450  Ord word 6361  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by: (None)