Theorem elirrvALT 9517

Description: Alternate proof of elirrv 9502, 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 9516	. 2 ⊢ E Fr 𝑥
2		efrirr 5598	. 2 ⊢ ( E Fr 𝑥 → ¬ 𝑥 ∈ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   E cep 5517   Fr wfr 5568

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 5218  ax-pr 5362  ax-reg 9497

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  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-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-eprel 5518  df-fr 5571

This theorem is referenced by: (None)