Theorem inf1 9215

Description: Variation of Axiom of Infinity (using zfinf 9232 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)

Hypothesis

Ref	Expression
inf1.1	⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Assertion

Ref	Expression
inf1	⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Proof of Theorem inf1

Step	Hyp	Ref	Expression
1		inf1.1	. 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
2		ne0i 4235	. . 3 ⊢ (𝑦 ∈ 𝑥 → 𝑥 ≠ ∅)
3	2	anim1i 618	. 2 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) → (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
4	1, 3	eximii 1844	1 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541  ∃wex 1787   ≠ wne 2932  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-dif 3856  df-nul 4224

This theorem is referenced by:  inf2  9216