Theorem inf2 3459

Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using axinf 1084 as a hypothesis).

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(s):   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 3458	. 2 ⊢ ∃x(¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))
3		dfss2 1497	. . . . 5 ⊢ (x ⊆ ∪x ↔ ∀y(y ∈ x → y ∈ ∪x))
4		eluni 1922	. . . . . . 7 ⊢ (y ∈ ∪x ↔ ∃z(y ∈ z ∧ z ∈ x))
5	4	imbi2i 160	. . . . . 6 ⊢ ((y ∈ x → y ∈ ∪x) ↔ (y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))
6	5	bial 695	. . . . 5 ⊢ (∀y(y ∈ x → y ∈ ∪x) ↔ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))
7	3, 6	bitr 151	. . . 4 ⊢ (x ⊆ ∪x ↔ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))
8	7	anbi2i 367	. . 3 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) ↔ (¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
9	8	biex 733	. 2 ⊢ (∃x(¬ x = ∅ ∧ x ⊆ ∪x) ↔ ∃x(¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
10	2, 9	mpbir 165	1 ⊢ ∃x(¬ x = ∅ ∧ x ⊆ ∪x)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919

This theorem is referenced by:  inf4 3473

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-uni 1920

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