Theorem axregszf 35266

Description: Derivation of zfregs 9645 using ax-regs 35263. (Contributed by BTernaryTau, 30-Dec-2025.)

Assertion

Ref	Expression
axregszf	⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem axregszf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4306	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		axregscl 35265	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
3		disj1 4405	. . . . 5 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
4	3	rexbii 3084	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
5		df-rex 3062	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
6	4, 5	bitr2i 276	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)) ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
7	2, 6	sylib 218	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
8	1, 7	sylbi 217	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061   ∩ cin 3901  ∅c0 4286

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-regs 35263

This theorem depends on definitions:  df-bi 207  df-an 396  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 3062  df-dif 3905  df-in 3909  df-nul 4287

This theorem is referenced by:  setindregs  35267  noinfepfnregs  35269