Theorem axunndlem1 9296

Description: Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
axunndlem1	⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Proof of Theorem axunndlem1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en2lp 8393	. . . . . . . 8 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
2		elequ2 1991	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
3	2	anbi2d 736	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
4	1, 3	mtbii 315	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
5	4	sps 2043	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
6	5	nexdv 1851	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
7	6	pm2.21d 117	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	axc4i 2116	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9		19.8a 2039	. . 3 ⊢ (∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
10	8, 9	syl 17	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
11		zfun 6848	. . 3 ⊢ ∃𝑥∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)
12		nfnae 2306	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
13		nfnae 2306	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
14		nfvd 1831	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 ∈ 𝑥)
15		nfcvf 2774	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
16	15	nfcrd 2757	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑧)
17	14, 16	nfand 1814	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
18	13, 17	nfexd 2153	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
19	18, 14	nfimd 1812	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
20		elequ1 1984	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
21	20	anbi1d 737	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
22	21	exbidv 1837	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
23	22, 20	imbi12d 333	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
24	23	a1i 11	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → ((∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
25	12, 19, 24	cbvald 2265	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
26	25	exbidv 1837	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
27	11, 26	mpbii 222	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
28	10, 27	pm2.61i 175	1 ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997

This theorem is referenced by:  axunnd  9297