Theorem axpownd 10620

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 2377. (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 10619	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
2		axpowndlem1 10616	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
3	2	aecoms 2433	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
4	2	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
5		nfnae 2439	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6		nfae 2438	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 𝑦 = 𝑧
7	5, 6	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8		el 5417	. . . . . . . . . . . . 13 ⊢ ∃𝑤 𝑥 ∈ 𝑤
9		nfcvf2 2927	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
10		nfcvd 2900	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
11	9, 10	nfeld 2911	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
12		elequ2 2124	. . . . . . . . . . . . . . 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 2413	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥 ∈ 𝑤 ↔ ∃𝑦 𝑥 ∈ 𝑦))
15	8, 14	mpbii 233	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥 ∈ 𝑦)
16	15	19.8ad 2183	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦)
17		df-ex 1780	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
18	16, 17	sylib 218	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
20		biidd 262	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))
21	20	dral1 2444	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ∀𝑧 ¬ 𝑥 ∈ 𝑦))
22		alnex 1781	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 𝑥 ∈ 𝑦)
23		alnex 1781	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
24	21, 22, 23	3bitr3g 313	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦))
25		nd2 10607	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧)
26		mtt 364	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑥 ∈ 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
28	24, 27	bitrd 279	. . . . . . . . . . 11 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
29	28	dral2 2443	. . . . . . . . . 10 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
30	29	adantl 481	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
31	19, 30	mtbid 324	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))
32	31	pm2.21d 121	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
33	7, 32	alrimi 2214	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
34	33	19.8ad 2183	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
35	34	a1d 25	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
36	35	ex 412	. . 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 182	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
38	1, 3, 37	pm2.61ii 183	1 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-reg 9611

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-pw 4582  df-sn 4607  df-pr 4609

This theorem is referenced by:  zfcndpow  10635  axpowprim  35726