Theorem axunndlem1 10020

Description: Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 9072	. . . . . . . 8 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
2		elequ2 2128	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
3	2	anbi2d 630	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
4	1, 3	mtbii 328	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
5	4	sps 2183	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
6	5	nexdv 1936	. . . . 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 2340	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	8	19.8ad 2180	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
10		zfun 7465	. . 3 ⊢ ∃𝑥∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)
11		nfnae 2455	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
12		nfnae 2455	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13		nfvd 1915	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 ∈ 𝑥)
14		nfcvf 3010	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
15	14	nfcrd 2972	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑧)
16	13, 15	nfand 1897	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
17	12, 16	nfexd 2347	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
18	17, 13	nfimd 1894	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
19		elequ1 2120	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
20	19	anbi1d 631	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
21	20	exbidv 1921	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
22	21, 19	imbi12d 347	. . . . . 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 2427	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑤(∃𝑥(𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
25	24	exbidv 1921	. . 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 235	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
27	9, 26	pm2.61i 184	1 ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464  ax-reg 9059

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-eprel 5468  df-fr 5517

This theorem is referenced by:  axunnd  10021