Theorem axunndlem1 10664

Description: Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

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 9675	. . . . . . . 8 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
2		elequ2 2123	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
3	2	anbi2d 629	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
4	1, 3	mtbii 326	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
5	4	sps 2186	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
6	5	nexdv 1935	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
7	6	pm2.21d 121	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	axc4i 2326	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	8	19.8ad 2183	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
10		zfun 7771	. . 3 ⊢ ∃𝑥∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)
11		nfnae 2442	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
12		nfnae 2442	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13		nfvd 1914	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 ∈ 𝑥)
14		nfcvf 2938	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
15	14	nfcrd 2902	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑧)
16	13, 15	nfand 1896	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
17	12, 16	nfexd 2333	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
18	17, 13	nfimd 1893	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
19		elequ1 2115	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
20	19	anbi1d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
21	20	exbidv 1920	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
22	21, 19	imbi12d 344	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
23	22	a1i 11	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → ((∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
24	11, 18, 23	cbvald 2415	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
25	24	exbidv 1920	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑥∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
26	10, 25	mpbii 233	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
27	9, 26	pm2.61i 182	1 ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-fr 5652

This theorem is referenced by:  axunnd  10665