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Theorem inf2 8558
 Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8574 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 8557 . 2 𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
3 dfss2 3624 . . . . 5 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 𝑥))
4 eluni 4471 . . . . . . 7 (𝑦 𝑥 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
54imbi2i 325 . . . . . 6 ((𝑦𝑥𝑦 𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
65albii 1787 . . . . 5 (∀𝑦(𝑦𝑥𝑦 𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
73, 6bitri 264 . . . 4 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 730 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1814 . 2 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
102, 9mpbir 221 1 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-uni 4469 This theorem is referenced by:  axinf2  8575
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