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Theorem vdwlem12 16864
Description: Lemma for vdw 16866. 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 14256 . . . . . . 7 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
31, 2syl 17 . . . . . 6 (𝜑 → (♯‘𝑅) ∈ ℕ0)
43nn0red 12474 . . . . 5 (𝜑 → (♯‘𝑅) ∈ ℝ)
54ltp1d 12085 . . . 4 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6 nn0p1nn 12452 . . . . . . 7 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
73, 6syl 17 . . . . . 6 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
87nnnn0d 12473 . . . . 5 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ0)
9 hashfz1 14246 . . . . 5 (((♯‘𝑅) + 1) ∈ ℕ0 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
108, 9syl 17 . . . 4 (𝜑 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
115, 10breqtrrd 5133 . . 3 (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12 fzfi 13877 . . . 4 (1...((♯‘𝑅) + 1)) ∈ Fin
13 hashsdom 14281 . . . 4 ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
141, 12, 13sylancl 586 . . 3 (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
1511, 14mpbid 231 . 2 (𝜑𝑅 ≺ (1...((♯‘𝑅) + 1)))
16 vdwlem12.f . . . . 5 (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17 fveq2 6842 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
18 fveq2 6842 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1917, 18eqeqan12d 2750 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
20 eqeq12 2753 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑧 = 𝑤𝑥 = 𝑦))
2119, 20imbi12d 344 . . . . . . 7 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
22 fveq2 6842 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
23 fveq2 6842 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2422, 23eqeqan12d 2750 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
25 eqcom 2743 . . . . . . . . 9 ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑥) = (𝐹𝑦))
2624, 25bitrdi 286 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
27 eqeq12 2753 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑦 = 𝑥))
28 eqcom 2743 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
2927, 28bitrdi 286 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑥 = 𝑦))
3026, 29imbi12d 344 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
31 elfznn 13470 . . . . . . . . . 10 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
3231nnred 12168 . . . . . . . . 9 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
3332ssriv 3948 . . . . . . . 8 (1...((♯‘𝑅) + 1)) ⊆ ℝ
3433a1i 11 . . . . . . 7 (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35 biidd 261 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
36 simplr3 1217 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥𝑦)
37 vdwlem12.2 . . . . . . . . . . 11 (𝜑 → ¬ 2 MonoAP 𝐹)
3837ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 2 MonoAP 𝐹)
39 3simpa 1148 . . . . . . . . . . . 12 ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40 simplrl 775 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43 simplrr 776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44 elfznn 13470 . . . . . . . . . . . . . . . . . 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 12197 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4741, 45, 46syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4842, 47mpbid 231 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦𝑥) ∈ ℕ)
49 df-2 12216 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
5049fveq2i 6845 . . . . . . . . . . . . . . . . . 18 (AP‘2) = (AP‘(1 + 1))
5150oveqi 7370 . . . . . . . . . . . . . . . . 17 (𝑥(AP‘2)(𝑦𝑥)) = (𝑥(AP‘(1 + 1))(𝑦𝑥))
52 1nn0 12429 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ0
53 vdwapun 16846 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℕ0𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5452, 41, 48, 53mp3an2i 1466 . . . . . . . . . . . . . . . . 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, 54eqtrid 2788 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) = (𝐹𝑦))
5716ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
5857ffnd 6669 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59 fniniseg 7010 . . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (𝐹 “ {(𝐹𝑦)}))
6261snssd 4769 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (𝐹 “ {(𝐹𝑦)}))
6341nncnd 12169 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
6445nncnd 12169 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
6563, 64pncan3d 11515 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦𝑥)) = 𝑦)
6665oveq1d 7372 . . . . . . . . . . . . . . . . . . 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 16849 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6845, 48, 67syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6966, 68eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = {𝑦})
70 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) = (𝐹𝑦))
71 fniniseg 7010 . . . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (𝐹 “ {(𝐹𝑦)}))
7473snssd 4769 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (𝐹 “ {(𝐹𝑦)}))
7569, 74eqsstrd 3982 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
7662, 75unssd 4146 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))) ⊆ (𝐹 “ {(𝐹𝑦)}))
7755, 76eqsstrd 3982 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
78 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
7978sseq1d 3975 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
80 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑦𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦𝑥)))
8180sseq1d 3975 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑦𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})))
8279, 81rspc2ev 3592 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . 15 (𝐹𝑦) ∈ V
85 sneq 4596 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝐹𝑦) → {𝑐} = {(𝐹𝑦)})
8685imaeq2d 6013 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝐹𝑦) → (𝐹 “ {𝑐}) = (𝐹 “ {(𝐹𝑦)}))
8786sseq2d 3976 . . . . . . . . . . . . . . . 16 (𝑐 = (𝐹𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
88872rexbidv 3213 . . . . . . . . . . . . . . 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 3565 . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . 14 (1...((♯‘𝑅) + 1)) ∈ V
92 2nn0 12430 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
9392a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ0)
9491, 93, 57vdwmc 16850 . . . . . . . . . . . . 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 256 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9639, 95sylanl2 679 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9796expr 457 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
9838, 97mtod 197 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝑥 < 𝑦)
99 simplr1 1215 . . . . . . . . . . 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 1216 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
10233, 101sselid 3942 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ ℝ)
103100, 102eqleltd 11299 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 = 𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑥 < 𝑦)))
10436, 98, 103mpbir2and 711 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
105104ex 413 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
10621, 30, 34, 35, 105wlogle 11688 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
107106ralrimivva 3197 . . . . 5 (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
108 dff13 7202 . . . . 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 583 . . . 4 (𝜑𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅)
110 f1domg 8912 . . . 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 9043 . . 3 ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
113111, 112syl 17 . 2 (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
11415, 113pm2.65i 193 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  cun 3908  wss 3910  {csn 4586   class class class wbr 5105  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  cdom 8881  csdm 8882  Fincfn 8883  cr 11050  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  ...cfz 13424  chash 14230  APcvdwa 16837   MonoAP cvdwm 16838
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-vdwap 16840  df-vdwmc 16841
This theorem is referenced by:  vdwlem13  16865
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