Theorem inf2 9642

Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9658 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(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Assertion

Ref	Expression
inf2	⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem inf2

Step	Hyp	Ref	Expression
1		inf1.1	. . 3 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
2	1	inf1 9641	. 2 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
3		df-ss 3948	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥))
4		eluni 4891	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
5	4	imbi2i 336	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
6	5	albii 1819	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
7	3, 6	bitri 275	. . . 4 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
8	7	anbi2i 623	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
9	8	exbii 1848	. 2 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
10	2, 9	mpbir 231	1 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933   ⊆ wss 3931  ∅c0 4313  ∪ cuni 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-v 3466  df-dif 3934  df-ss 3948  df-nul 4314  df-uni 4889

This theorem is referenced by:  axinf2  9659