Metamath Proof Explorer


Theorem axnulALT2

Description: Alternate proof of axnul , proved from propositional calculus, ax-gen , ax-4 , ax-5 , and ax-inf2 . (Contributed by BTernaryTau, 22-Jun-2025) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axnulALT2 𝑥𝑦 ¬ 𝑦𝑥

Proof

Step Hyp Ref Expression
1 exsimpr ( ∃ 𝑥 ( 𝑥𝑧 ∧ ∀ 𝑦 ¬ 𝑦𝑥 ) → ∃ 𝑥𝑦 ¬ 𝑦𝑥 )
2 ax-inf2 𝑧 ( ∃ 𝑥 ( 𝑥𝑧 ∧ ∀ 𝑦 ¬ 𝑦𝑥 ) ∧ ∀ 𝑥 ( 𝑥𝑧 → ∃ 𝑦 ( 𝑦𝑧 ∧ ∀ 𝑤 ( 𝑤𝑦 ↔ ( 𝑤𝑥𝑤 = 𝑥 ) ) ) ) )
3 simpl ( ( ∃ 𝑥 ( 𝑥𝑧 ∧ ∀ 𝑦 ¬ 𝑦𝑥 ) ∧ ∀ 𝑥 ( 𝑥𝑧 → ∃ 𝑦 ( 𝑦𝑧 ∧ ∀ 𝑤 ( 𝑤𝑦 ↔ ( 𝑤𝑥𝑤 = 𝑥 ) ) ) ) ) → ∃ 𝑥 ( 𝑥𝑧 ∧ ∀ 𝑦 ¬ 𝑦𝑥 ) )
4 2 3 eximii 𝑧𝑥 ( 𝑥𝑧 ∧ ∀ 𝑦 ¬ 𝑦𝑥 )
5 1 4 exlimiiv 𝑥𝑦 ¬ 𝑦𝑥