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Theorem disjxpin 30004
Description: Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
Hypotheses
Ref Expression
disjxpin.1 (𝑥 = (1st𝑝) → 𝐶 = 𝐸)
disjxpin.2 (𝑦 = (2nd𝑝) → 𝐷 = 𝐹)
disjxpin.3 (𝜑Disj 𝑥𝐴 𝐶)
disjxpin.4 (𝜑Disj 𝑦𝐵 𝐷)
Assertion
Ref Expression
disjxpin (𝜑Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸𝐹))
Distinct variable groups:   𝑥,𝑝,𝐴   𝑦,𝑝,𝐵   𝐶,𝑝   𝐷,𝑝   𝑥,𝐸   𝑦,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑦,𝑝)   𝐹(𝑥,𝑝)

Proof of Theorem disjxpin
Dummy variables 𝑎 𝑐 𝑞 𝑟 𝑏 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7568 . . . . . . . . 9 (𝑞 ∈ (𝐴 × 𝐵) → (1st𝑞) ∈ 𝐴)
21ad2antrl 724 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (1st𝑞) ∈ 𝐴)
3 xp1st 7568 . . . . . . . . 9 (𝑟 ∈ (𝐴 × 𝐵) → (1st𝑟) ∈ 𝐴)
43ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (1st𝑟) ∈ 𝐴)
5 simpl 483 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → 𝜑)
6 disjxpin.3 . . . . . . . . . . 11 (𝜑Disj 𝑥𝐴 𝐶)
7 disjors 4939 . . . . . . . . . . 11 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑎𝐴𝑐𝐴 (𝑎 = 𝑐 ∨ (𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅))
86, 7sylib 219 . . . . . . . . . 10 (𝜑 → ∀𝑎𝐴𝑐𝐴 (𝑎 = 𝑐 ∨ (𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅))
9 eqeq1 2797 . . . . . . . . . . . 12 (𝑎 = (1st𝑞) → (𝑎 = 𝑐 ↔ (1st𝑞) = 𝑐))
10 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑎 = (1st𝑞) → 𝑎 / 𝑥𝐶 = (1st𝑞) / 𝑥𝐶)
1110ineq1d 4103 . . . . . . . . . . . . 13 (𝑎 = (1st𝑞) → (𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶))
1211eqeq1d 2795 . . . . . . . . . . . 12 (𝑎 = (1st𝑞) → ((𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅ ↔ ((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅))
139, 12orbi12d 911 . . . . . . . . . . 11 (𝑎 = (1st𝑞) → ((𝑎 = 𝑐 ∨ (𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅) ↔ ((1st𝑞) = 𝑐 ∨ ((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅)))
14 eqeq2 2804 . . . . . . . . . . . 12 (𝑐 = (1st𝑟) → ((1st𝑞) = 𝑐 ↔ (1st𝑞) = (1st𝑟)))
15 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑐 = (1st𝑟) → 𝑐 / 𝑥𝐶 = (1st𝑟) / 𝑥𝐶)
1615ineq2d 4104 . . . . . . . . . . . . 13 (𝑐 = (1st𝑟) → ((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶) = ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶))
1716eqeq1d 2795 . . . . . . . . . . . 12 (𝑐 = (1st𝑟) → (((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅ ↔ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅))
1814, 17orbi12d 911 . . . . . . . . . . 11 (𝑐 = (1st𝑟) → (((1st𝑞) = 𝑐 ∨ ((1st𝑞) / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅) ↔ ((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅)))
1913, 18rspc2v 3567 . . . . . . . . . 10 (((1st𝑞) ∈ 𝐴 ∧ (1st𝑟) ∈ 𝐴) → (∀𝑎𝐴𝑐𝐴 (𝑎 = 𝑐 ∨ (𝑎 / 𝑥𝐶𝑐 / 𝑥𝐶) = ∅) → ((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅)))
208, 19syl5 34 . . . . . . . . 9 (((1st𝑞) ∈ 𝐴 ∧ (1st𝑟) ∈ 𝐴) → (𝜑 → ((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅)))
2120imp 407 . . . . . . . 8 ((((1st𝑞) ∈ 𝐴 ∧ (1st𝑟) ∈ 𝐴) ∧ 𝜑) → ((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅))
222, 4, 5, 21syl21anc 834 . . . . . . 7 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅))
23 xp2nd 7569 . . . . . . . . 9 (𝑞 ∈ (𝐴 × 𝐵) → (2nd𝑞) ∈ 𝐵)
2423ad2antrl 724 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (2nd𝑞) ∈ 𝐵)
25 xp2nd 7569 . . . . . . . . 9 (𝑟 ∈ (𝐴 × 𝐵) → (2nd𝑟) ∈ 𝐵)
2625ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (2nd𝑟) ∈ 𝐵)
27 disjxpin.4 . . . . . . . . . . 11 (𝜑Disj 𝑦𝐵 𝐷)
28 disjors 4939 . . . . . . . . . . 11 (Disj 𝑦𝐵 𝐷 ↔ ∀𝑏𝐵𝑑𝐵 (𝑏 = 𝑑 ∨ (𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅))
2927, 28sylib 219 . . . . . . . . . 10 (𝜑 → ∀𝑏𝐵𝑑𝐵 (𝑏 = 𝑑 ∨ (𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅))
30 eqeq1 2797 . . . . . . . . . . . 12 (𝑏 = (2nd𝑞) → (𝑏 = 𝑑 ↔ (2nd𝑞) = 𝑑))
31 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑏 = (2nd𝑞) → 𝑏 / 𝑦𝐷 = (2nd𝑞) / 𝑦𝐷)
3231ineq1d 4103 . . . . . . . . . . . . 13 (𝑏 = (2nd𝑞) → (𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷))
3332eqeq1d 2795 . . . . . . . . . . . 12 (𝑏 = (2nd𝑞) → ((𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅ ↔ ((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅))
3430, 33orbi12d 911 . . . . . . . . . . 11 (𝑏 = (2nd𝑞) → ((𝑏 = 𝑑 ∨ (𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅) ↔ ((2nd𝑞) = 𝑑 ∨ ((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅)))
35 eqeq2 2804 . . . . . . . . . . . 12 (𝑑 = (2nd𝑟) → ((2nd𝑞) = 𝑑 ↔ (2nd𝑞) = (2nd𝑟)))
36 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑑 = (2nd𝑟) → 𝑑 / 𝑦𝐷 = (2nd𝑟) / 𝑦𝐷)
3736ineq2d 4104 . . . . . . . . . . . . 13 (𝑑 = (2nd𝑟) → ((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷) = ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷))
3837eqeq1d 2795 . . . . . . . . . . . 12 (𝑑 = (2nd𝑟) → (((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅ ↔ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))
3935, 38orbi12d 911 . . . . . . . . . . 11 (𝑑 = (2nd𝑟) → (((2nd𝑞) = 𝑑 ∨ ((2nd𝑞) / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅) ↔ ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))
4034, 39rspc2v 3567 . . . . . . . . . 10 (((2nd𝑞) ∈ 𝐵 ∧ (2nd𝑟) ∈ 𝐵) → (∀𝑏𝐵𝑑𝐵 (𝑏 = 𝑑 ∨ (𝑏 / 𝑦𝐷𝑑 / 𝑦𝐷) = ∅) → ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))
4129, 40syl5 34 . . . . . . . . 9 (((2nd𝑞) ∈ 𝐵 ∧ (2nd𝑟) ∈ 𝐵) → (𝜑 → ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))
4241imp 407 . . . . . . . 8 ((((2nd𝑞) ∈ 𝐵 ∧ (2nd𝑟) ∈ 𝐵) ∧ 𝜑) → ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))
4324, 26, 5, 42syl21anc 834 . . . . . . 7 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))
4422, 43jca 512 . . . . . 6 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅) ∧ ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))
45 anddi 1003 . . . . . 6 ((((1st𝑞) = (1st𝑟) ∨ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅) ∧ ((2nd𝑞) = (2nd𝑟) ∨ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)) ↔ ((((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ ((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))))
4644, 45sylib 219 . . . . 5 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ ((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))))
47 orass 914 . . . . 5 (((((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ ((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))) ↔ (((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))))
4846, 47sylib 219 . . . 4 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))))
49 xpopth 7577 . . . . . . 7 ((𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵)) → (((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ↔ 𝑞 = 𝑟))
5049adantl 482 . . . . . 6 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ↔ 𝑞 = 𝑟))
5150biimpd 230 . . . . 5 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) → 𝑞 = 𝑟))
52 inss2 4121 . . . . . . . . . 10 ((𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸) ∩ (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹)) ⊆ (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹)
53 csbin 4300 . . . . . . . . . . . 12 𝑞 / 𝑝(𝐸𝐹) = (𝑞 / 𝑝𝐸𝑞 / 𝑝𝐹)
54 csbin 4300 . . . . . . . . . . . 12 𝑟 / 𝑝(𝐸𝐹) = (𝑟 / 𝑝𝐸𝑟 / 𝑝𝐹)
5553, 54ineq12i 4102 . . . . . . . . . . 11 (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ((𝑞 / 𝑝𝐸𝑞 / 𝑝𝐹) ∩ (𝑟 / 𝑝𝐸𝑟 / 𝑝𝐹))
56 in4 4117 . . . . . . . . . . 11 ((𝑞 / 𝑝𝐸𝑞 / 𝑝𝐹) ∩ (𝑟 / 𝑝𝐸𝑟 / 𝑝𝐹)) = ((𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸) ∩ (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹))
5755, 56eqtri 2817 . . . . . . . . . 10 (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ((𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸) ∩ (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹))
58 vex 3435 . . . . . . . . . . . . 13 𝑞 ∈ V
59 csbnestg 4287 . . . . . . . . . . . . 13 (𝑞 ∈ V → 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷)
6058, 59ax-mp 5 . . . . . . . . . . . 12 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷
61 fvex 6543 . . . . . . . . . . . . . 14 (2nd𝑝) ∈ V
62 disjxpin.2 . . . . . . . . . . . . . 14 (𝑦 = (2nd𝑝) → 𝐷 = 𝐹)
6361, 62csbie 3838 . . . . . . . . . . . . 13 (2nd𝑝) / 𝑦𝐷 = 𝐹
6463csbeq2i 3814 . . . . . . . . . . . 12 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑞 / 𝑝𝐹
65 csbfv 6575 . . . . . . . . . . . . 13 𝑞 / 𝑝(2nd𝑝) = (2nd𝑞)
66 csbeq1 3809 . . . . . . . . . . . . 13 (𝑞 / 𝑝(2nd𝑝) = (2nd𝑞) → 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷 = (2nd𝑞) / 𝑦𝐷)
6765, 66ax-mp 5 . . . . . . . . . . . 12 𝑞 / 𝑝(2nd𝑝) / 𝑦𝐷 = (2nd𝑞) / 𝑦𝐷
6860, 64, 673eqtr3ri 2826 . . . . . . . . . . 11 (2nd𝑞) / 𝑦𝐷 = 𝑞 / 𝑝𝐹
69 vex 3435 . . . . . . . . . . . . 13 𝑟 ∈ V
70 csbnestg 4287 . . . . . . . . . . . . 13 (𝑟 ∈ V → 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷)
7169, 70ax-mp 5 . . . . . . . . . . . 12 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷
7263csbeq2i 3814 . . . . . . . . . . . 12 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷 = 𝑟 / 𝑝𝐹
73 csbfv 6575 . . . . . . . . . . . . 13 𝑟 / 𝑝(2nd𝑝) = (2nd𝑟)
74 csbeq1 3809 . . . . . . . . . . . . 13 (𝑟 / 𝑝(2nd𝑝) = (2nd𝑟) → 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷 = (2nd𝑟) / 𝑦𝐷)
7573, 74ax-mp 5 . . . . . . . . . . . 12 𝑟 / 𝑝(2nd𝑝) / 𝑦𝐷 = (2nd𝑟) / 𝑦𝐷
7671, 72, 753eqtr3ri 2826 . . . . . . . . . . 11 (2nd𝑟) / 𝑦𝐷 = 𝑟 / 𝑝𝐹
7768, 76ineq12i 4102 . . . . . . . . . 10 ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹)
7852, 57, 773sstr4i 3926 . . . . . . . . 9 (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) ⊆ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷)
79 sseq0 4267 . . . . . . . . 9 (((𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) ⊆ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅)
8078, 79mpan 686 . . . . . . . 8 (((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅ → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅)
8180a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅ → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
8281adantld 491 . . . . . 6 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
83 inss1 4120 . . . . . . . . . . 11 ((𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸) ∩ (𝑞 / 𝑝𝐹𝑟 / 𝑝𝐹)) ⊆ (𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸)
84 csbnestg 4287 . . . . . . . . . . . . . 14 (𝑞 ∈ V → 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶)
8558, 84ax-mp 5 . . . . . . . . . . . . 13 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶
86 fvex 6543 . . . . . . . . . . . . . . 15 (1st𝑝) ∈ V
87 disjxpin.1 . . . . . . . . . . . . . . 15 (𝑥 = (1st𝑝) → 𝐶 = 𝐸)
8886, 87csbie 3838 . . . . . . . . . . . . . 14 (1st𝑝) / 𝑥𝐶 = 𝐸
8988csbeq2i 3814 . . . . . . . . . . . . 13 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑞 / 𝑝𝐸
90 csbfv 6575 . . . . . . . . . . . . . 14 𝑞 / 𝑝(1st𝑝) = (1st𝑞)
91 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑞 / 𝑝(1st𝑝) = (1st𝑞) → 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶 = (1st𝑞) / 𝑥𝐶)
9290, 91ax-mp 5 . . . . . . . . . . . . 13 𝑞 / 𝑝(1st𝑝) / 𝑥𝐶 = (1st𝑞) / 𝑥𝐶
9385, 89, 923eqtr3ri 2826 . . . . . . . . . . . 12 (1st𝑞) / 𝑥𝐶 = 𝑞 / 𝑝𝐸
94 csbnestg 4287 . . . . . . . . . . . . . 14 (𝑟 ∈ V → 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶)
9569, 94ax-mp 5 . . . . . . . . . . . . 13 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶
9688csbeq2i 3814 . . . . . . . . . . . . 13 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶 = 𝑟 / 𝑝𝐸
97 csbfv 6575 . . . . . . . . . . . . . 14 𝑟 / 𝑝(1st𝑝) = (1st𝑟)
98 csbeq1 3809 . . . . . . . . . . . . . 14 (𝑟 / 𝑝(1st𝑝) = (1st𝑟) → 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶 = (1st𝑟) / 𝑥𝐶)
9997, 98ax-mp 5 . . . . . . . . . . . . 13 𝑟 / 𝑝(1st𝑝) / 𝑥𝐶 = (1st𝑟) / 𝑥𝐶
10095, 96, 993eqtr3ri 2826 . . . . . . . . . . . 12 (1st𝑟) / 𝑥𝐶 = 𝑟 / 𝑝𝐸
10193, 100ineq12i 4102 . . . . . . . . . . 11 ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = (𝑞 / 𝑝𝐸𝑟 / 𝑝𝐸)
10283, 57, 1013sstr4i 3926 . . . . . . . . . 10 (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) ⊆ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶)
103 sseq0 4267 . . . . . . . . . 10 (((𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) ⊆ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) ∧ ((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅)
104102, 103mpan 686 . . . . . . . . 9 (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅)
105104a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
106105adantrd 492 . . . . . . 7 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
10781adantld 491 . . . . . . 7 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
108106, 107jaod 854 . . . . . 6 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
10982, 108jaod 854 . . . . 5 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅))) → (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
11051, 109orim12d 957 . . . 4 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → ((((1st𝑞) = (1st𝑟) ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) = (1st𝑟) ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅) ∨ ((((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ (2nd𝑞) = (2nd𝑟)) ∨ (((1st𝑞) / 𝑥𝐶(1st𝑟) / 𝑥𝐶) = ∅ ∧ ((2nd𝑞) / 𝑦𝐷(2nd𝑟) / 𝑦𝐷) = ∅)))) → (𝑞 = 𝑟 ∨ (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅)))
11148, 110mpd 15 . . 3 ((𝜑 ∧ (𝑞 ∈ (𝐴 × 𝐵) ∧ 𝑟 ∈ (𝐴 × 𝐵))) → (𝑞 = 𝑟 ∨ (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
112111ralrimivva 3156 . 2 (𝜑 → ∀𝑞 ∈ (𝐴 × 𝐵)∀𝑟 ∈ (𝐴 × 𝐵)(𝑞 = 𝑟 ∨ (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
113 disjors 4939 . 2 (Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸𝐹) ↔ ∀𝑞 ∈ (𝐴 × 𝐵)∀𝑟 ∈ (𝐴 × 𝐵)(𝑞 = 𝑟 ∨ (𝑞 / 𝑝(𝐸𝐹) ∩ 𝑟 / 𝑝(𝐸𝐹)) = ∅))
114112, 113sylibr 235 1 (𝜑Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 842   = wceq 1520  wcel 2079  wral 3103  Vcvv 3432  csb 3806  cin 3853  wss 3854  c0 4206  Disj wdisj 4924   × cxp 5433  cfv 6217  1st c1st 7534  2nd c2nd 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-disj 4925  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-iota 6181  df-fun 6219  df-fv 6225  df-1st 7536  df-2nd 7537
This theorem is referenced by:  sibfof  31171
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