Theorem zfcndinf 10042

Description: Axiom of Infinity ax-inf 9103, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 5272 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 5272	. . 3 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 ∈ 𝑦
3		nfe1 2154	. . . . . . . 8 ⊢ Ⅎ𝑤∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
4	2, 3	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
5	4	nfal 2342	. . . . . 6 ⊢ Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
6	2, 5	nfan 1900	. . . . 5 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
7	6	nfex 2343	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
8		axinfnd 10030	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑤 → (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
9	8	19.37iv 1949	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
10	7, 9	exlimi 2217	. . 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 631	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
15	14	exbidv 1922	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
16	12, 15	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
17	16	cbvalvw 2043	. . . 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 1848	. 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 233	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-reg 9058  ax-inf 9103

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-nul 4294  df-sn 4570  df-pr 4572

This theorem is referenced by: (None)