Theorem onprcALT 36632

Description: Alternate proof of onprc 7816 from onprcr 36631. (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 6384	. . 3 ⊢ On = {𝑥 ∣ Ord 𝑥}
2		rabab 3484	. . 3 ⊢ {𝑥 ∈ V ∣ Ord 𝑥} = {𝑥 ∣ Ord 𝑥}
3	1, 2	eqtr4i 2789	. 2 ⊢ On = {𝑥 ∈ V ∣ Ord 𝑥}
4		tron 6403	. 2 ⊢ Tr On
5	3, 4	onprcr 36631	1 ⊢ ¬ On ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  {cab 2741  {crab 3416  Vcvv 3454  Ord word 6379  Oncon0 6380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383  df-on 6384

This theorem is used by: (None)