Theorem zfcndinf 10541

Description: Axiom of Infinity ax-inf 9559, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5394 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)

Assertion

Ref	Expression
zfcndinf	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndinf

Step	Hyp	Ref	Expression
1		el 5394	. . 3 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 ∈ 𝑦
3		nfe1 2156	. . . . . . . 8 ⊢ Ⅎ𝑤∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
4	2, 3	nfim 1898	. . . . . . 7 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
5	4	nfal 2329	. . . . . 6 ⊢ Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
6	2, 5	nfan 1901	. . . . 5 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
7	6	nfex 2330	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
8		axinfnd 10529	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑤 → (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
9	8	19.37iv 1950	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
10	7, 9	exlimi 2225	. . 3 ⊢ (∃𝑤 𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
11	1, 10	ax-mp 5	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
12		elequ1 2121	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
13		elequ1 2121	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑥 ∈ 𝑤))
14	13	anbi1d 632	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
15	14	exbidv 1923	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
16	12, 15	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
17	16	cbvalvw 2038	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
18	17	anbi2i 624	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
19	18	exbii 1850	. 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
20	11, 19	mpbir 231	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-reg 9509  ax-inf 9559

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by: (None)