Theorem elirrvALT 9613

Description: Alternate proof of elirrv 9598, shorter but using more axioms. (Contributed by BTernaryTau, 28-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elirrvALT	⊢ ¬ 𝑥 ∈ 𝑥

Proof of Theorem elirrvALT

Step	Hyp	Ref	Expression
1		zfregfr 9612	. 2 ⊢ E Fr 𝑥
2		efrirr 5631	. 2 ⊢ ( E Fr 𝑥 → ¬ 𝑥 ∈ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝑥 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   E cep 5550   Fr wfr 5601

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 5391  ax-reg 9593

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-eprel 5551  df-fr 5604

This theorem is used by: (None)