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Theorem inf2 9074
 Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9090 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
Hypothesis
Ref Expression
inf1.1 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
Assertion
Ref Expression
inf2 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem inf2
StepHypRef Expression
1 inf1.1 . . 3 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
21inf1 9073 . 2 𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
3 dfss2 3904 . . . . 5 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 𝑥))
4 eluni 4806 . . . . . . 7 (𝑦 𝑥 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
54imbi2i 339 . . . . . 6 ((𝑦𝑥𝑦 𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
65albii 1821 . . . . 5 (∀𝑦(𝑦𝑥𝑦 𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
73, 6bitri 278 . . . 4 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 625 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1849 . 2 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
102, 9mpbir 234 1 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990   ⊆ wss 3884  ∅c0 4246  ∪ cuni 4803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-uni 4804 This theorem is referenced by:  axinf2  9091
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