Theorem axregszf 35063

Description: Derivation of zfregs 9647 using ax-regs 35060. (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 35062	. . 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 3076	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
5		df-rex 3054	. . . 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∩ cin 3904  ∅c0 4286

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-ext 2701  ax-regs 35060

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-dif 3908  df-in 3912  df-nul 4287

This theorem is referenced by:  setindregs  35064