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Theorem fphpdo 39929
 Description: Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
Hypotheses
Ref Expression
fphpdo.1 (𝜑𝐴 ⊆ ℝ)
fphpdo.2 (𝜑𝐵 ∈ V)
fphpdo.3 (𝜑𝐵𝐴)
fphpdo.4 ((𝜑𝑧𝐴) → 𝐶𝐵)
fphpdo.5 (𝑧 = 𝑥𝐶 = 𝐷)
fphpdo.6 (𝑧 = 𝑦𝐶 = 𝐸)
Assertion
Ref Expression
fphpdo (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷,𝑧   𝑥,𝐸,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑧)   𝐷(𝑥)   𝐸(𝑦)

Proof of Theorem fphpdo
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fphpdo.3 . . 3 (𝜑𝐵𝐴)
2 fphpdo.4 . . . . 5 ((𝜑𝑧𝐴) → 𝐶𝐵)
32fmpttd 6866 . . . 4 (𝜑 → (𝑧𝐴𝐶):𝐴𝐵)
43ffvelrnda 6838 . . 3 ((𝜑𝑏𝐴) → ((𝑧𝐴𝐶)‘𝑏) ∈ 𝐵)
5 fveq2 6655 . . 3 (𝑏 = 𝑐 → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
61, 4, 5fphpd 39928 . 2 (𝜑 → ∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
7 fphpdo.1 . . . . . . . . . 10 (𝜑𝐴 ⊆ ℝ)
87sselda 3917 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑏 ∈ ℝ)
98adantrr 716 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏 ∈ ℝ)
109adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑏 ∈ ℝ)
117sselda 3917 . . . . . . . . 9 ((𝜑𝑐𝐴) → 𝑐 ∈ ℝ)
1211adantrl 715 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐 ∈ ℝ)
1312adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑐 ∈ ℝ)
1410, 13lttri2d 10786 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 ↔ (𝑏 < 𝑐𝑐 < 𝑏)))
15 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏𝐴)
1615ad2antrr 725 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏𝐴)
17 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐𝐴)
1817ad2antrr 725 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑐𝐴)
19 simpr 488 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏 < 𝑐)
20 simplr 768 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
21 breq1 5037 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (𝑥 < 𝑦𝑏 < 𝑦))
22 fveqeq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)))
2321, 22anbi12d 633 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦))))
24 breq2 5038 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝑏 < 𝑦𝑏 < 𝑐))
25 fveq2 6655 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑐))
2625eqeq2d 2809 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
2724, 26anbi12d 633 . . . . . . . . . . 11 (𝑦 = 𝑐 → ((𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))))
2823, 27rspc2ev 3584 . . . . . . . . . 10 ((𝑏𝐴𝑐𝐴 ∧ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
2916, 18, 19, 20, 28syl112anc 1371 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3029ex 416 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏 < 𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
3117ad2antrr 725 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐𝐴)
3215ad2antrr 725 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑏𝐴)
33 simpr 488 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐 < 𝑏)
34 simplr 768 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
3534eqcomd 2804 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))
36 breq1 5037 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (𝑥 < 𝑦𝑐 < 𝑦))
37 fveqeq2 6664 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)))
3836, 37anbi12d 633 . . . . . . . . . . 11 (𝑥 = 𝑐 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦))))
39 breq2 5038 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑐 < 𝑦𝑐 < 𝑏))
40 fveq2 6655 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑏))
4140eqeq2d 2809 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏)))
4239, 41anbi12d 633 . . . . . . . . . . 11 (𝑦 = 𝑏 → ((𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))))
4338, 42rspc2ev 3584 . . . . . . . . . 10 ((𝑐𝐴𝑏𝐴 ∧ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4431, 32, 33, 35, 43syl112anc 1371 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4544ex 416 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑐 < 𝑏 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
4630, 45jaod 856 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
47 eqid 2798 . . . . . . . . . . . . . 14 (𝑧𝐴𝐶) = (𝑧𝐴𝐶)
48 fphpdo.5 . . . . . . . . . . . . . 14 (𝑧 = 𝑥𝐶 = 𝐷)
49 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
50 eleq1w 2872 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
5150anbi2d 631 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝜑𝑧𝐴) ↔ (𝜑𝑥𝐴)))
5248eleq1d 2874 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝐶𝐵𝐷𝐵))
5351, 52imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑥𝐴) → 𝐷𝐵)))
5453, 2chvarvv 2005 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐷𝐵)
5554adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐷𝐵)
5647, 48, 49, 55fvmptd3 6778 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
57 fphpdo.6 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝐶 = 𝐸)
58 simpr 488 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
59 eleq1w 2872 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6059anbi2d 631 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝜑𝑧𝐴) ↔ (𝜑𝑦𝐴)))
6157eleq1d 2874 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝐶𝐵𝐸𝐵))
6260, 61imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑦𝐴) → 𝐸𝐵)))
6362, 2chvarvv 2005 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → 𝐸𝐵)
6463adantlr 714 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐸𝐵)
6547, 57, 58, 64fvmptd3 6778 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
6656, 65eqeq12d 2814 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ 𝐷 = 𝐸))
6766biimpd 232 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) → 𝐷 = 𝐸))
6867anim2d 614 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → (𝑥 < 𝑦𝐷 = 𝐸)))
6968reximdva 3234 . . . . . . . . 9 ((𝜑𝑥𝐴) → (∃𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7069reximdva 3234 . . . . . . . 8 (𝜑 → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7170ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7246, 71syld 47 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7314, 72sylbid 243 . . . . 5 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7473expimpd 457 . . . 4 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐) ∧ 𝑏𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7574ancomsd 469 . . 3 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7675rexlimdvva 3254 . 2 (𝜑 → (∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
776, 76mpd 15 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  Vcvv 3442   ⊆ wss 3883   class class class wbr 5034   ↦ cmpt 5114  ‘cfv 6332   ≺ csdm 8509  ℝcr 10543   < clt 10682 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-resscn 10601  ax-pre-lttri 10618  ax-pre-lttrn 10619 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-ltxr 10687 This theorem is referenced by:  irrapxlem1  39934
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