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Theorem inf2 9567
Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9583 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 9566 . 2 𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
3 dfss2 3934 . . . . 5 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 𝑥))
4 eluni 4872 . . . . . . 7 (𝑦 𝑥 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
54imbi2i 336 . . . . . 6 ((𝑦𝑥𝑦 𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
65albii 1822 . . . . 5 (∀𝑦(𝑦𝑥𝑦 𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
73, 6bitri 275 . . . 4 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 624 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1851 . 2 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
102, 9mpbir 230 1 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wex 1782  wcel 2107  wne 2940  wss 3914  c0 4286   cuni 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-uni 4870
This theorem is referenced by:  axinf2  9584
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