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Theorem vdwlem12 16545
Description: Lemma for vdw 16547. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdwlem12.f (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
vdwlem12.2 (𝜑 → ¬ 2 MonoAP 𝐹)
Assertion
Ref Expression
vdwlem12 ¬ 𝜑

Proof of Theorem vdwlem12
Dummy variables 𝑎 𝑐 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw.r . . . . . . 7 (𝜑𝑅 ∈ Fin)
2 hashcl 13923 . . . . . . 7 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
31, 2syl 17 . . . . . 6 (𝜑 → (♯‘𝑅) ∈ ℕ0)
43nn0red 12151 . . . . 5 (𝜑 → (♯‘𝑅) ∈ ℝ)
54ltp1d 11762 . . . 4 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6 nn0p1nn 12129 . . . . . . 7 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
73, 6syl 17 . . . . . 6 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
87nnnn0d 12150 . . . . 5 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ0)
9 hashfz1 13912 . . . . 5 (((♯‘𝑅) + 1) ∈ ℕ0 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
108, 9syl 17 . . . 4 (𝜑 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
115, 10breqtrrd 5081 . . 3 (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12 fzfi 13545 . . . 4 (1...((♯‘𝑅) + 1)) ∈ Fin
13 hashsdom 13948 . . . 4 ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
141, 12, 13sylancl 589 . . 3 (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
1511, 14mpbid 235 . 2 (𝜑𝑅 ≺ (1...((♯‘𝑅) + 1)))
16 vdwlem12.f . . . . 5 (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17 fveq2 6717 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
18 fveq2 6717 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1917, 18eqeqan12d 2751 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
20 eqeq12 2754 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑧 = 𝑤𝑥 = 𝑦))
2119, 20imbi12d 348 . . . . . . 7 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
22 fveq2 6717 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
23 fveq2 6717 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2422, 23eqeqan12d 2751 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
25 eqcom 2744 . . . . . . . . 9 ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑥) = (𝐹𝑦))
2624, 25bitrdi 290 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
27 eqeq12 2754 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑦 = 𝑥))
28 eqcom 2744 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
2927, 28bitrdi 290 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑥 = 𝑦))
3026, 29imbi12d 348 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
31 elfznn 13141 . . . . . . . . . 10 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
3231nnred 11845 . . . . . . . . 9 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
3332ssriv 3905 . . . . . . . 8 (1...((♯‘𝑅) + 1)) ⊆ ℝ
3433a1i 11 . . . . . . 7 (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35 biidd 265 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
36 simplr3 1219 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥𝑦)
37 vdwlem12.2 . . . . . . . . . . 11 (𝜑 → ¬ 2 MonoAP 𝐹)
3837ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 2 MonoAP 𝐹)
39 3simpa 1150 . . . . . . . . . . . 12 ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40 simplrl 777 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
4140, 31syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℕ)
42 simprr 773 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43 simplrr 778 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44 elfznn 13141 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...((♯‘𝑅) + 1)) → 𝑦 ∈ ℕ)
4543, 44syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℕ)
46 nnsub 11874 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4741, 45, 46syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4842, 47mpbid 235 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦𝑥) ∈ ℕ)
49 df-2 11893 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
5049fveq2i 6720 . . . . . . . . . . . . . . . . . 18 (AP‘2) = (AP‘(1 + 1))
5150oveqi 7226 . . . . . . . . . . . . . . . . 17 (𝑥(AP‘2)(𝑦𝑥)) = (𝑥(AP‘(1 + 1))(𝑦𝑥))
52 1nn0 12106 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ0
53 vdwapun 16527 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℕ0𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5452, 41, 48, 53mp3an2i 1468 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5551, 54syl5eq 2790 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
56 simprl 771 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) = (𝐹𝑦))
5716ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
5857ffnd 6546 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59 fniniseg 6880 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6058, 59syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6140, 56, 60mpbir2and 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (𝐹 “ {(𝐹𝑦)}))
6261snssd 4722 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (𝐹 “ {(𝐹𝑦)}))
6341nncnd 11846 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
6445nncnd 11846 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
6563, 64pncan3d 11192 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦𝑥)) = 𝑦)
6665oveq1d 7228 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = (𝑦(AP‘1)(𝑦𝑥)))
67 vdwap1 16530 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6845, 48, 67syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6966, 68eqtrd 2777 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = {𝑦})
70 eqidd 2738 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) = (𝐹𝑦))
71 fniniseg 6880 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7258, 71syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7343, 70, 72mpbir2and 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (𝐹 “ {(𝐹𝑦)}))
7473snssd 4722 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (𝐹 “ {(𝐹𝑦)}))
7569, 74eqsstrd 3939 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
7662, 75unssd 4100 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))) ⊆ (𝐹 “ {(𝐹𝑦)}))
7755, 76eqsstrd 3939 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
78 oveq1 7220 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
7978sseq1d 3932 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
80 oveq2 7221 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑦𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦𝑥)))
8180sseq1d 3932 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑦𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})))
8279, 81rspc2ev 3549 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ ∧ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
8341, 48, 77, 82syl3anc 1373 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
84 fvex 6730 . . . . . . . . . . . . . . 15 (𝐹𝑦) ∈ V
85 sneq 4551 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝐹𝑦) → {𝑐} = {(𝐹𝑦)})
8685imaeq2d 5929 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝐹𝑦) → (𝐹 “ {𝑐}) = (𝐹 “ {(𝐹𝑦)}))
8786sseq2d 3933 . . . . . . . . . . . . . . . 16 (𝑐 = (𝐹𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
88872rexbidv 3219 . . . . . . . . . . . . . . 15 (𝑐 = (𝐹𝑦) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
8984, 88spcev 3521 . . . . . . . . . . . . . 14 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
9083, 89syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
91 ovex 7246 . . . . . . . . . . . . . 14 (1...((♯‘𝑅) + 1)) ∈ V
92 2nn0 12107 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
9392a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ0)
9491, 93, 57vdwmc 16531 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (2 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐})))
9590, 94mpbird 260 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9639, 95sylanl2 681 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9796expr 460 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
9838, 97mtod 201 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝑥 < 𝑦)
99 simplr1 1217 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
10099, 32syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ ℝ)
101 simplr2 1218 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
10233, 101sselid 3898 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ ℝ)
103100, 102eqleltd 10976 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 = 𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑥 < 𝑦)))
10436, 98, 103mpbir2and 713 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
105104ex 416 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
10621, 30, 34, 35, 105wlogle 11365 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
107106ralrimivva 3112 . . . . 5 (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
108 dff13 7067 . . . . 5 (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 ↔ (𝐹:(1...((♯‘𝑅) + 1))⟶𝑅 ∧ ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
10916, 107, 108sylanbrc 586 . . . 4 (𝜑𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅)
110 f1domg 8648 . . . 4 (𝑅 ∈ Fin → (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 → (1...((♯‘𝑅) + 1)) ≼ 𝑅))
1111, 109, 110sylc 65 . . 3 (𝜑 → (1...((♯‘𝑅) + 1)) ≼ 𝑅)
112 domnsym 8772 . . 3 ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
113111, 112syl 17 . 2 (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
11415, 113pm2.65i 197 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  wral 3061  wrex 3062  cun 3864  wss 3866  {csn 4541   class class class wbr 5053  ccnv 5550  cima 5554   Fn wfn 6375  wf 6376  1-1wf1 6377  cfv 6380  (class class class)co 7213  cdom 8624  csdm 8625  Fincfn 8626  cr 10728  1c1 10730   + caddc 10732   < clt 10867  cle 10868  cmin 11062  cn 11830  2c2 11885  0cn0 12090  ...cfz 13095  chash 13896  APcvdwa 16518   MonoAP cvdwm 16519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-hash 13897  df-vdwap 16521  df-vdwmc 16522
This theorem is referenced by:  vdwlem13  16546
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