Theorem axregszf 35642

Description: Derivation of zfregs 9714 using ax-regs 35639. (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 4303	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		axregscl 35641	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
3		disj1 4408	. . . . 5 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
4	3	rexbii 3111	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
5		df-rex 3089	. . . 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 2957  ∃wrex 3088   ∩ cin 3901  ∅c0 4282

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 2734  ax-regs 35639

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 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-dif 3905  df-in 3909  df-nul 4283

This theorem is used by:  setindregs  35643  noinfepfnregs  35645