Theorem inf1 3458

Description: Variation of Axiom of Infinity (using axinf 1084 as a hypothesis). Axiom of Infinity of [FreydScedrov] p. 283.

Hypothesis

Ref	Expression
inf1.1	⊢ ∃x(y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))

Assertion

Ref	Expression
inf1	⊢ ∃x(¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))

Distinct variable group(s):   x,y,z

Proof of Theorem inf1

Step	Hyp	Ref	Expression
1		inf1.1	. 2 ⊢ ∃x(y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))
2		n0i 1712	. . . 4 ⊢ (y ∈ x → ¬ x = ∅)
3	2	anim1i 269	. . 3 ⊢ ((y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))) → (¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
4	3	19.22i 723	. 2 ⊢ (∃x(y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))) → ∃x(¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
5	1, 4	ax-mp 6	1 ⊢ ∃x(¬ x = ∅ ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   = wceq 1091  ∅c0 1707

This theorem is referenced by:  inf2 3459

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-nul 1708

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