Theorem axregszf 35385

Description: Derivation of zfregs 9682 using ax-regs 35382. (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 4305	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		axregscl 35384	. . 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 3108	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴))
5		df-rex 3086	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)))
6	4, 5	bitr2i 278	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴)) ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
7	2, 6	sylib 220	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
8	1, 7	sylbi 219	1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085   ∩ cin 3903  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-regs 35382

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-dif 3907  df-in 3911  df-nul 4286

This theorem is referenced by:  setindregs  35386  noinfepfnregs  35388