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Theorem inf2 8464
Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8480 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 8463 . 2 𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
3 dfss2 3572 . . . . 5 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 𝑥))
4 eluni 4405 . . . . . . 7 (𝑦 𝑥 ↔ ∃𝑧(𝑦𝑧𝑧𝑥))
54imbi2i 326 . . . . . 6 ((𝑦𝑥𝑦 𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
65albii 1744 . . . . 5 (∀𝑦(𝑦𝑥𝑦 𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
73, 6bitri 264 . . . 4 (𝑥 𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 729 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1771 . 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 384  wal 1478  wex 1701  wcel 1987  wne 2790  wss 3555  c0 3891   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-uni 4403
This theorem is referenced by:  axinf2  8481
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