Theorem epfrs 9489

Description: The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9490. (Contributed by Mario Carneiro, 22-Mar-2013.)

Assertion

Ref	Expression
epfrs	⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem epfrs

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4280	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
2		snssi 4741	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐴 → {𝑧} ⊆ 𝐴)
3	2	anim2i 617	. . . . . . . . . . 11 ⊢ (({𝑧} ⊆ 𝑦 ∧ 𝑧 ∈ 𝐴) → ({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴))
4		ssin 4164	. . . . . . . . . . . 12 ⊢ (({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴) ↔ {𝑧} ⊆ (𝑦 ∩ 𝐴))
5		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
6	5	snss 4719	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∩ 𝐴) ↔ {𝑧} ⊆ (𝑦 ∩ 𝐴))
7	4, 6	bitr4i 277	. . . . . . . . . . 11 ⊢ (({𝑧} ⊆ 𝑦 ∧ {𝑧} ⊆ 𝐴) ↔ 𝑧 ∈ (𝑦 ∩ 𝐴))
8	3, 7	sylib 217	. . . . . . . . . 10 ⊢ (({𝑧} ⊆ 𝑦 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑦 ∩ 𝐴))
9	8	ne0d 4269	. . . . . . . . 9 ⊢ (({𝑧} ⊆ 𝑦 ∧ 𝑧 ∈ 𝐴) → (𝑦 ∩ 𝐴) ≠ ∅)
10		inss2 4163	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
11		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
12	11	inex1 5241	. . . . . . . . . . . . 13 ⊢ (𝑦 ∩ 𝐴) ∈ V
13	12	epfrc 5575	. . . . . . . . . . . 12 ⊢ (( E Fr 𝐴 ∧ (𝑦 ∩ 𝐴) ⊆ 𝐴 ∧ (𝑦 ∩ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∩ 𝐴)((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅)
14	10, 13	mp3an2 1448	. . . . . . . . . . 11 ⊢ (( E Fr 𝐴 ∧ (𝑦 ∩ 𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦 ∩ 𝐴)((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅)
15		elin 3903	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑦 ∩ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴))
16	15	anbi1i 624	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑦 ∩ 𝐴) ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅))
17		anass 469	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅) ↔ (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝐴 ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅)))
18	16, 17	bitri 274	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑦 ∩ 𝐴) ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅) ↔ (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝐴 ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅)))
19		n0 4280	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∩ 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑥 ∩ 𝐴))
20		elinel1 4129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ (𝑥 ∩ 𝐴) → 𝑤 ∈ 𝑥)
21	20	ancri 550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝑥 ∩ 𝐴) → (𝑤 ∈ 𝑥 ∧ 𝑤 ∈ (𝑥 ∩ 𝐴)))
22		trel 5198	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Tr 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) → 𝑤 ∈ 𝑦))
23		inass 4153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝑥) = (𝑦 ∩ (𝐴 ∩ 𝑥))
24		incom 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐴 ∩ 𝑥) = (𝑥 ∩ 𝐴)
25	24	ineq2i 4143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∩ (𝐴 ∩ 𝑥)) = (𝑦 ∩ (𝑥 ∩ 𝐴))
26	23, 25	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝑥) = (𝑦 ∩ (𝑥 ∩ 𝐴))
27	26	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ ((𝑦 ∩ 𝐴) ∩ 𝑥) ↔ 𝑤 ∈ (𝑦 ∩ (𝑥 ∩ 𝐴)))
28		elin 3903	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ (𝑦 ∩ (𝑥 ∩ 𝐴)) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ (𝑥 ∩ 𝐴)))
29	27, 28	bitr2i 275	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑤 ∈ (𝑥 ∩ 𝐴)) ↔ 𝑤 ∈ ((𝑦 ∩ 𝐴) ∩ 𝑥))
30		ne0i 4268	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 ∈ ((𝑦 ∩ 𝐴) ∩ 𝑥) → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)
31	29, 30	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑤 ∈ (𝑥 ∩ 𝐴)) → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)
32	31	ex 413	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 ∈ 𝑦 → (𝑤 ∈ (𝑥 ∩ 𝐴) → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅))
33	22, 32	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Tr 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) → (𝑤 ∈ (𝑥 ∩ 𝐴) → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
34	33	expd 416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Tr 𝑦 → (𝑤 ∈ 𝑥 → (𝑥 ∈ 𝑦 → (𝑤 ∈ (𝑥 ∩ 𝐴) → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅))))
35	34	com34 91	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Tr 𝑦 → (𝑤 ∈ 𝑥 → (𝑤 ∈ (𝑥 ∩ 𝐴) → (𝑥 ∈ 𝑦 → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅))))
36	35	impd 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Tr 𝑦 → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ (𝑥 ∩ 𝐴)) → (𝑥 ∈ 𝑦 → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
37	21, 36	syl5 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr 𝑦 → (𝑤 ∈ (𝑥 ∩ 𝐴) → (𝑥 ∈ 𝑦 → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
38	37	exlimdv 1936	. . . . . . . . . . . . . . . . . . 19 ⊢ (Tr 𝑦 → (∃𝑤 𝑤 ∈ (𝑥 ∩ 𝐴) → (𝑥 ∈ 𝑦 → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
39	19, 38	syl5bi 241	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑦 → ((𝑥 ∩ 𝐴) ≠ ∅ → (𝑥 ∈ 𝑦 → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
40	39	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (Tr 𝑦 → (𝑥 ∈ 𝑦 → ((𝑥 ∩ 𝐴) ≠ ∅ → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅)))
41	40	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ 𝑦) → ((𝑥 ∩ 𝐴) ≠ ∅ → ((𝑦 ∩ 𝐴) ∩ 𝑥) ≠ ∅))
42	41	necon4d 2967	. . . . . . . . . . . . . . 15 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ 𝑦) → (((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅ → (𝑥 ∩ 𝐴) = ∅))
43	42	anim2d 612	. . . . . . . . . . . . . 14 ⊢ ((Tr 𝑦 ∧ 𝑥 ∈ 𝑦) → ((𝑥 ∈ 𝐴 ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∩ 𝐴) = ∅)))
44	43	expimpd 454	. . . . . . . . . . . . 13 ⊢ (Tr 𝑦 → ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝐴 ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅)) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∩ 𝐴) = ∅)))
45	18, 44	syl5bi 241	. . . . . . . . . . . 12 ⊢ (Tr 𝑦 → ((𝑥 ∈ (𝑦 ∩ 𝐴) ∧ ((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅) → (𝑥 ∈ 𝐴 ∧ (𝑥 ∩ 𝐴) = ∅)))
46	45	reximdv2 3199	. . . . . . . . . . 11 ⊢ (Tr 𝑦 → (∃𝑥 ∈ (𝑦 ∩ 𝐴)((𝑦 ∩ 𝐴) ∩ 𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))
47	14, 46	syl5 34	. . . . . . . . . 10 ⊢ (Tr 𝑦 → (( E Fr 𝐴 ∧ (𝑦 ∩ 𝐴) ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))
48	47	expcomd 417	. . . . . . . . 9 ⊢ (Tr 𝑦 → ((𝑦 ∩ 𝐴) ≠ ∅ → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)))
49	9, 48	syl5 34	. . . . . . . 8 ⊢ (Tr 𝑦 → (({𝑧} ⊆ 𝑦 ∧ 𝑧 ∈ 𝐴) → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)))
50	49	expd 416	. . . . . . 7 ⊢ (Tr 𝑦 → ({𝑧} ⊆ 𝑦 → (𝑧 ∈ 𝐴 → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))))
51	50	impcom 408	. . . . . 6 ⊢ (({𝑧} ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∈ 𝐴 → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)))
52	51	3adant3 1131	. . . . 5 ⊢ (({𝑧} ⊆ 𝑦 ∧ Tr 𝑦 ∧ ∀𝑤(({𝑧} ⊆ 𝑤 ∧ Tr 𝑤) → 𝑦 ⊆ 𝑤)) → (𝑧 ∈ 𝐴 → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)))
53		snex 5354	. . . . . 6 ⊢ {𝑧} ∈ V
54	53	tz9.1 9487	. . . . 5 ⊢ ∃𝑦({𝑧} ⊆ 𝑦 ∧ Tr 𝑦 ∧ ∀𝑤(({𝑧} ⊆ 𝑤 ∧ Tr 𝑤) → 𝑦 ⊆ 𝑤))
55	52, 54	exlimiiv 1934	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))
56	55	exlimiv 1933	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))
57	1, 56	sylbi 216	. 2 ⊢ (𝐴 ≠ ∅ → ( E Fr 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅))
58	57	impcom 408	1 ⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  Tr wtr 5191   E cep 5494   Fr wfr 5541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241

This theorem is referenced by:  zfregs  9490