Theorem inf1 9665

Description: Variation of Axiom of Infinity (using zfinf 9682 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 4337	. . 3 ⊢ (𝑦 ∈ 𝑥 → 𝑥 ≠ ∅)
3	2	anim1i 613	. 2 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) → (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
4	1, 3	eximii 1832	1 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1532  ∃wex 1774   ≠ wne 2930  ∅c0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-dif 3950  df-nul 4326

This theorem is referenced by:  inf2  9666