Theorem axregszf 35722

Description: Derivation of zfregs 9711 using ax-regs 35719. (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 4299	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		axregscl 35721	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
3		disj1 4404	. . . . 5 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
4	3	rexbii 3109	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
5		df-rex 3087	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
6	4, 5	bitr2i 279	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)) ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
7	2, 6	sylib 221	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
8	1, 7	sylbi 220	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   ∩ cin 3897  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-regs 35719

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-dif 3901  df-in 3905  df-nul 4279

This theorem is used by:  setindregs  35723  noinfepfnregs  35725