Theorem elirrvALT 9536

Description: Alternate proof of elirrv 9525, 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 9535	. 2 ⊢ E Fr 𝑥
2		efrirr 5611	. 2 ⊢ ( E Fr 𝑥 → ¬ 𝑥 ∈ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   E cep 5530   Fr wfr 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-reg 9521

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-eprel 5531  df-fr 5584

This theorem is referenced by: (None)