Theorem axpownd 10022

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 2386. (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 10021	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
2		axpowndlem1 10018	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
3	2	aecoms 2446	. 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 2452	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6		nfae 2451	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 𝑦 = 𝑧
7	5, 6	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧)
8		el 5269	. . . . . . . . . . . . 13 ⊢ ∃𝑤 𝑥 ∈ 𝑤
9		nfcvf2 3008	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
10		nfcvd 2978	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
11	9, 10	nfeld 2989	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
12		elequ2 2125	. . . . . . . . . . . . . . 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 2425	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤 𝑥 ∈ 𝑤 ↔ ∃𝑦 𝑥 ∈ 𝑦))
15	8, 14	mpbii 235	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦 𝑥 ∈ 𝑦)
16	15	19.8ad 2177	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦 𝑥 ∈ 𝑦)
17		df-ex 1777	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
18	16, 17	sylib 220	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
19	18	adantr 483	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ¬ ∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦)
20		biidd 264	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))
21	20	dral1 2457	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ∀𝑧 ¬ 𝑥 ∈ 𝑦))
22		alnex 1778	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 𝑥 ∈ 𝑦)
23		alnex 1778	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
24	21, 22, 23	3bitr3g 315	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦))
25		nd2 10009	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
28	24, 27	bitrd 281	. . . . . . . . . . 11 ⊢ (∀𝑦 𝑦 = 𝑧 → (¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
29	28	dral2 2456	. . . . . . . . . 10 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
30	29	adantl 484	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 ¬ ∃𝑦 𝑥 ∈ 𝑦 ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
31	19, 30	mtbid 326	. . . . . . . 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 2209	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
34	33	19.8ad 2177	. . . . 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 415	. . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-reg 9055

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4567  df-pr 4569

This theorem is referenced by:  zfcndpow  10037  axpowprim  32930