Theorem onfrALTlem2 43297

Description: Lemma for onfrALT 43300. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem2	⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))

Distinct variable groups:   𝑦,𝑎   𝑥,𝑦

Proof of Theorem onfrALTlem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑦))
2	1	2a1i 12	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑦))))
3		inss2 4229	. . . . . . . . . . . 12 ⊢ (𝑎 ∩ 𝑦) ⊆ 𝑦
4	3	sseli 3978	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ 𝑦)
5	2, 4	syl8 76	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑦)))
6		inss1 4228	. . . . . . . . . . . . 13 ⊢ (𝑎 ∩ 𝑦) ⊆ 𝑎
7	6	sseli 3978	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ 𝑎)
8	2, 7	syl8 76	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑎)))
9		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ On)
10		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → 𝑥 ∈ 𝑎)
11		ssel 3975	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ⊆ On → (𝑥 ∈ 𝑎 → 𝑥 ∈ On))
12	9, 10, 11	syl2im 40	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → 𝑥 ∈ On))
13		eloni 6374	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → Ord 𝑥)
14	12, 13	syl6 35	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → Ord 𝑥))
15		ordtr 6378	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → Tr 𝑥)
16	14, 15	syl6 35	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → Tr 𝑥))
17		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ (𝑎 ∩ 𝑥))
18	17	2a1i 12	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ (𝑎 ∩ 𝑥))))
19		inss2 4229	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∩ 𝑥) ⊆ 𝑥
20	19	sseli 3978	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑎 ∩ 𝑥) → 𝑦 ∈ 𝑥)
21	18, 20	syl8 76	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑦 ∈ 𝑥)))
22		trel 5274	. . . . . . . . . . . . 13 ⊢ (Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
23	22	expcomd 417	. . . . . . . . . . . 12 ⊢ (Tr 𝑥 → (𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥)))
24	16, 21, 5, 23	ee233 43270	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ 𝑥)))
25		elin 3964	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑎 ∩ 𝑥) ↔ (𝑧 ∈ 𝑎 ∧ 𝑧 ∈ 𝑥))
26	25	simplbi2 501	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑎 → (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝑎 ∩ 𝑥)))
27	8, 24, 26	ee33 43272	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ (𝑎 ∩ 𝑥))))
28		elin 3964	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦) ↔ (𝑧 ∈ (𝑎 ∩ 𝑥) ∧ 𝑧 ∈ 𝑦))
29	28	simplbi2com 503	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ (𝑎 ∩ 𝑥) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)))
30	5, 27, 29	ee33 43272	. . . . . . . . 9 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → (((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) ∧ 𝑧 ∈ (𝑎 ∩ 𝑦)) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦))))
31	30	exp4a 432	. . . . . . . 8 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)))))
32	31	ggen31 43296	. . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)))))
33		dfss2 3968	. . . . . . . 8 ⊢ ((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) ↔ ∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)))
34	33	biimpri 227	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ (𝑎 ∩ 𝑦) → 𝑧 ∈ ((𝑎 ∩ 𝑥) ∩ 𝑦)) → (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦))
35	32, 34	syl8 76	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦))))
36		simpr 485	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)
37	36	2a1i 12	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)))
38		sseq0 4399	. . . . . . 7 ⊢ (((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) = ∅)
39	38	ex 413	. . . . . 6 ⊢ ((𝑎 ∩ 𝑦) ⊆ ((𝑎 ∩ 𝑥) ∩ 𝑦) → (((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ → (𝑎 ∩ 𝑦) = ∅))
40	35, 37, 39	ee33 43272	. . . . 5 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑎 ∩ 𝑦) = ∅)))
41		simpl 483	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎 ∩ 𝑥))
42	41	2a1i 12	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ (𝑎 ∩ 𝑥))))
43		inss1 4228	. . . . . . 7 ⊢ (𝑎 ∩ 𝑥) ⊆ 𝑎
44	43	sseli 3978	. . . . . 6 ⊢ (𝑦 ∈ (𝑎 ∩ 𝑥) → 𝑦 ∈ 𝑎)
45	42, 44	syl8 76	. . . . 5 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝑎)))
46		pm3.21 472	. . . . 5 ⊢ ((𝑎 ∩ 𝑦) = ∅ → (𝑦 ∈ 𝑎 → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)))
47	40, 45, 46	ee33 43272	. . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))))
48	47	alrimdv 1932	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))))
49		onfrALTlem3 43295	. . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
50		df-rex 3071	. . . 4 ⊢ (∃𝑦 ∈ (𝑎 ∩ 𝑥)((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅))
51	49, 50	imbitrdi 250	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅)))
52		exim 1836	. . 3 ⊢ (∀𝑦((𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) → (∃𝑦(𝑦 ∈ (𝑎 ∩ 𝑥) ∧ ((𝑎 ∩ 𝑥) ∩ 𝑦) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)))
53	48, 51, 52	syl6c 70	. 2 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)))
54		df-rex 3071	. 2 ⊢ (∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
55	53, 54	syl6ibr 251	1 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∃wrex 3070   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  Tr wtr 5265  Ord word 6363  Oncon0 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368

This theorem is referenced by:  onfrALT  43300