Theorem elirrvALT 9520

Description: Alternate proof of elirrv 9506, 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 9519	. 2 ⊢ E Fr 𝑥
2		efrirr 5605	. 2 ⊢ ( E Fr 𝑥 → ¬ 𝑥 ∈ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   E cep 5524   Fr wfr 5575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5525  df-fr 5578

This theorem is referenced by: (None)