Theorem axpownd 10667

Description: A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
axpownd	⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

Proof of Theorem axpownd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axpowndlem4 10666	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
2		axpowndlem1 10663	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
3	2	aecoms 2458	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
4	2	a1d 26	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
5		nfnae 2464	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6		nfae 2463	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 𝑦 = 𝑧
7	5, 6	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8		el 5406	. . . . . . . . . . . . 13 ⊢ ∃𝑤 𝑥 ∈ 𝑤
9		nfcvf2 2950	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
10		nfcvd 2924	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
11	9, 10	nfeld 2934	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
12		elequ2 2160	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦)))
14	5, 11, 13	cbvexd 2438	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥 ∈ 𝑤 ↔ ∃𝑦 𝑥 ∈ 𝑦))
15	8, 14	mpbii 236	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥 ∈ 𝑦)
16	15	19.8ad 2219	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦)
17		df-ex 1813	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
18	16, 17	sylib 221	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
19	18	adantr 486	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
20		biidd 265	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))
21	20	dral1 2469	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ∀𝑧 ¬ 𝑥 ∈ 𝑦))
22		alnex 1814	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 𝑥 ∈ 𝑦)
23		alnex 1814	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
24	21, 22, 23	3bitr3g 316	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦))
25		nd2 10654	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧)
26		mtt 367	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑥 ∈ 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
27	25, 26	syl 18	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
28	24, 27	bitrd 282	. . . . . . . . . . 11 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
29	28	dral2 2468	. . . . . . . . . 10 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
30	29	adantl 487	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
31	19, 30	mtbid 327	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))
32	31	pm2.21d 122	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
33	7, 32	alrimi 2250	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
34	33	19.8ad 2219	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
35	34	a1d 26	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
36	35	ex 418	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
37	4, 36	pm2.61i 184	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
38	1, 3, 37	pm2.61ii 185	1 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2402  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-reg 9570

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3453  df-dif 3902  df-ss 3916  df-nul 4280  df-pw 4559  df-sn 4585

This theorem is used by:  zfcndpow  10682  axpowprim  36438