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Theorem ramub1lem1 16352
Description: Lemma for ramub1 16354. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ramub1.m (𝜑𝑀 ∈ ℕ)
ramub1.r (𝜑𝑅 ∈ Fin)
ramub1.f (𝜑𝐹:𝑅⟶ℕ)
ramub1.g 𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))
ramub1.1 (𝜑𝐺:𝑅⟶ℕ0)
ramub1.2 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)
ramub1.3 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
ramub1.4 (𝜑𝑆 ∈ Fin)
ramub1.5 (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
ramub1.6 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
ramub1.x (𝜑𝑋𝑆)
ramub1.h 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
ramub1.d (𝜑𝐷𝑅)
ramub1.w (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))
ramub1.7 (𝜑 → (𝐺𝐷) ≤ (♯‘𝑊))
ramub1.8 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
ramub1.e (𝜑𝐸𝑅)
ramub1.v (𝜑𝑉𝑊)
ramub1.9 (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
ramub1.s (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
Assertion
Ref Expression
ramub1lem1 (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
Distinct variable groups:   𝑥,𝑢,𝐷   𝑦,𝑢,𝑧,𝐹,𝑥   𝑎,𝑏,𝑖,𝑢,𝑥,𝑦,𝑧,𝑀   𝐺,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧   𝑢,𝑅,𝑥,𝑦,𝑧   𝑊,𝑎,𝑖,𝑢   𝜑,𝑢,𝑥,𝑦,𝑧   𝑆,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧   𝑉,𝑎,𝑖,𝑥,𝑧   𝑢,𝐶,𝑥,𝑦,𝑧   𝑢,𝐻,𝑥,𝑦,𝑧   𝑢,𝐾,𝑥,𝑦,𝑧   𝑥,𝐸,𝑧   𝑋,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝐷(𝑦,𝑧,𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝑆(𝑏)   𝐸(𝑦,𝑢,𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝐺(𝑏)   𝐻(𝑖,𝑎,𝑏)   𝐾(𝑖,𝑎,𝑏)   𝑉(𝑦,𝑢,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑏)   𝑋(𝑏)

Proof of Theorem ramub1lem1
StepHypRef Expression
1 ramub1.4 . . . . 5 (𝜑𝑆 ∈ Fin)
2 ramub1.v . . . . . . . 8 (𝜑𝑉𝑊)
3 ramub1.w . . . . . . . 8 (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))
42, 3sstrd 3981 . . . . . . 7 (𝜑𝑉 ⊆ (𝑆 ∖ {𝑋}))
54difss2d 4115 . . . . . 6 (𝜑𝑉𝑆)
6 ramub1.x . . . . . . 7 (𝜑𝑋𝑆)
76snssd 4741 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝑆)
85, 7unssd 4166 . . . . 5 (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
91, 8sselpwd 5227 . . . 4 (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
109adantr 481 . . 3 ((𝜑𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
11 iftrue 4476 . . . . . . 7 (𝐸 = 𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
1211adantl 482 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
13 ramub1.9 . . . . . . 7 (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
1413adantr 481 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
1512, 14eqbrtrrd 5087 . . . . 5 ((𝜑𝐸 = 𝐷) → ((𝐹𝐷) − 1) ≤ (♯‘𝑉))
16 ramub1.f . . . . . . . . 9 (𝜑𝐹:𝑅⟶ℕ)
17 ramub1.d . . . . . . . . 9 (𝜑𝐷𝑅)
1816, 17ffvelrnd 6848 . . . . . . . 8 (𝜑 → (𝐹𝐷) ∈ ℕ)
1918adantr 481 . . . . . . 7 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℕ)
2019nnred 11642 . . . . . 6 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℝ)
21 1red 10631 . . . . . 6 ((𝜑𝐸 = 𝐷) → 1 ∈ ℝ)
221, 5ssfid 8730 . . . . . . . 8 (𝜑𝑉 ∈ Fin)
23 hashcl 13707 . . . . . . . 8 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
24 nn0re 11895 . . . . . . . 8 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
2522, 23, 243syl 18 . . . . . . 7 (𝜑 → (♯‘𝑉) ∈ ℝ)
2625adantr 481 . . . . . 6 ((𝜑𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
2720, 21, 26lesubaddd 11226 . . . . 5 ((𝜑𝐸 = 𝐷) → (((𝐹𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
2815, 27mpbid 233 . . . 4 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ≤ ((♯‘𝑉) + 1))
29 fveq2 6667 . . . . 5 (𝐸 = 𝐷 → (𝐹𝐸) = (𝐹𝐷))
30 snidg 4596 . . . . . . . 8 (𝑋𝑆𝑋 ∈ {𝑋})
316, 30syl 17 . . . . . . 7 (𝜑𝑋 ∈ {𝑋})
324sseld 3970 . . . . . . . 8 (𝜑 → (𝑋𝑉𝑋 ∈ (𝑆 ∖ {𝑋})))
33 eldifn 4108 . . . . . . . 8 (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
3432, 33syl6 35 . . . . . . 7 (𝜑 → (𝑋𝑉 → ¬ 𝑋 ∈ {𝑋}))
3531, 34mt2d 138 . . . . . 6 (𝜑 → ¬ 𝑋𝑉)
36 hashunsng 13743 . . . . . . 7 (𝑋𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
376, 36syl 17 . . . . . 6 (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
3822, 35, 37mp2and 695 . . . . 5 (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
3929, 38breqan12rd 5080 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
4028, 39mpbird 258 . . 3 ((𝜑𝐸 = 𝐷) → (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})))
41 snfi 8583 . . . . . . 7 {𝑋} ∈ Fin
42 unfi 8774 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
4322, 41, 42sylancl 586 . . . . . 6 (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
44 ramub1.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
4544nnnn0d 11944 . . . . . 6 (𝜑𝑀 ∈ ℕ0)
46 ramub1.3 . . . . . . 7 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
4746hashbcval 16328 . . . . . 6 (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ0) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
4843, 45, 47syl2anc 584 . . . . 5 (𝜑 → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
4948adantr 481 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
50 simpl1l 1218 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
5146hashbcval 16328 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
5222, 45, 51syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
53 ramub1.s . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
5452, 53eqsstrrd 4010 . . . . . . . 8 (𝜑 → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
5550, 54syl 17 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
56 simpr 485 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑉)
57 simpl3 1187 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
58 rabid 3384 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (♯‘𝑥) = 𝑀))
5956, 57, 58sylanbrc 583 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
6055, 59sseldd 3972 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
61 simpl2 1186 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
6261elpwid 4556 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
63 simpl1l 1218 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
6463, 8syl 17 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
6562, 64sstrd 3981 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥𝑆)
66 vex 3503 . . . . . . . . . . 11 𝑥 ∈ V
6766elpw 4549 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑆𝑥𝑆)
6865, 67sylibr 235 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
69 simpl3 1187 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
70 rabid 3384 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑆 ∧ (♯‘𝑥) = 𝑀))
7168, 69, 70sylanbrc 583 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7246hashbcval 16328 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
731, 45, 72syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7463, 73syl 17 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7571, 74eleqtrrd 2921 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
763difss2d 4115 . . . . . . . . . . . . . . 15 (𝜑𝑊𝑆)
771, 76ssfid 8730 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ Fin)
78 nnm1nn0 11927 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
7944, 78syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8046hashbcval 16328 . . . . . . . . . . . . . 14 ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
8177, 79, 80syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
82 ramub1.8 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
8381, 82eqsstrrd 4010 . . . . . . . . . . . 12 (𝜑 → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (𝐻 “ {𝐷}))
8463, 83syl 17 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (𝐻 “ {𝐷}))
85 fveqeq2 6676 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
86 uncom 4133 . . . . . . . . . . . . . . . 16 (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
8762, 86sseqtrdi 4021 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
88 ssundif 4436 . . . . . . . . . . . . . . 15 (𝑥 ⊆ ({𝑋} ∪ 𝑉) ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑉)
8987, 88sylib 219 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑉)
9063, 2syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑉𝑊)
9189, 90sstrd 3981 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
9266difexi 5229 . . . . . . . . . . . . . 14 (𝑥 ∖ {𝑋}) ∈ V
9392elpw 4549 . . . . . . . . . . . . 13 ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑊)
9491, 93sylibr 235 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊)
9563, 1syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑆 ∈ Fin)
9695, 65ssfid 8730 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
97 diffi 8739 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin → (𝑥 ∖ {𝑋}) ∈ Fin)
9896, 97syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ Fin)
99 hashcl 13707 . . . . . . . . . . . . . . 15 ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
100 nn0cn 11896 . . . . . . . . . . . . . . 15 ((♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0 → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
10198, 99, 1003syl 18 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
102 ax-1cn 10584 . . . . . . . . . . . . . 14 1 ∈ ℂ
103 pncan 10881 . . . . . . . . . . . . . 14 (((♯‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
104101, 102, 103sylancl 586 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
105 neldifsnd 4725 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
106 hashunsng 13743 . . . . . . . . . . . . . . . . 17 (𝑋𝑆 → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
10763, 6, 1063syl 18 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
10898, 105, 107mp2and 695 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
109 undif1 4427 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∖ {𝑋}) ∪ {𝑋}) = (𝑥 ∪ {𝑋})
110 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑥 ∈ 𝒫 𝑉)
11161, 110eldifd 3951 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
112 elpwunsn 4620 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉) → 𝑋𝑥)
113111, 112syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑋𝑥)
114113snssd 4741 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
115 ssequn2 4163 . . . . . . . . . . . . . . . . . . 19 ({𝑋} ⊆ 𝑥 ↔ (𝑥 ∪ {𝑋}) = 𝑥)
116114, 115sylib 219 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∪ {𝑋}) = 𝑥)
117109, 116syl5req 2874 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
118117fveq2d 6671 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
119118, 69eqtr3d 2863 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
120108, 119eqtr3d 2863 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
121120oveq1d 7163 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
122104, 121eqtr3d 2863 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
12385, 94, 122elrabd 3686 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
12484, 123sseldd 3972 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ (𝐻 “ {𝐷}))
125 ramub1.h . . . . . . . . . . . 12 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
126125mptiniseg 6091 . . . . . . . . . . 11 (𝐷𝑅 → (𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
12763, 17, 1263syl 18 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
128124, 127eleqtrd 2920 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
129 uneq1 4136 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
130129fveqeq2d 6675 . . . . . . . . . . 11 (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
131130elrab 3684 . . . . . . . . . 10 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} ↔ ((𝑥 ∖ {𝑋}) ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∧ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
132131simprbi 497 . . . . . . . . 9 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
133128, 132syl 17 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
134117fveq2d 6671 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
135 simpl1r 1219 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
136133, 134, 1353eqtr4d 2871 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = 𝐸)
137 ramub1.6 . . . . . . . . 9 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
138137ffnd 6512 . . . . . . . 8 (𝜑𝐾 Fn (𝑆𝐶𝑀))
139 fniniseg 6826 . . . . . . . 8 (𝐾 Fn (𝑆𝐶𝑀) → (𝑥 ∈ (𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾𝑥) = 𝐸)))
14063, 138, 1393syl 18 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∈ (𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾𝑥) = 𝐸)))
14175, 136, 140mpbir2and 709 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
14260, 141pm2.61dan 809 . . . . 5 (((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) → 𝑥 ∈ (𝐾 “ {𝐸}))
143142rabssdv 4055 . . . 4 ((𝜑𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
14449, 143eqsstrd 4009 . . 3 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
145 fveq2 6667 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
146145breq2d 5075 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
147 oveq1 7155 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
148147sseq1d 4002 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
149146, 148anbi12d 630 . . . 4 (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
150149rspcev 3627 . . 3 (((𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
15110, 40, 144, 150syl12anc 834 . 2 ((𝜑𝐸 = 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
1521, 5sselpwd 5227 . . . 4 (𝜑𝑉 ∈ 𝒫 𝑆)
153152adantr 481 . . 3 ((𝜑𝐸𝐷) → 𝑉 ∈ 𝒫 𝑆)
154 ifnefalse 4482 . . . . 5 (𝐸𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
155154adantl 482 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
15613adantr 481 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
157155, 156eqbrtrrd 5087 . . 3 ((𝜑𝐸𝐷) → (𝐹𝐸) ≤ (♯‘𝑉))
15853adantr 481 . . 3 ((𝜑𝐸𝐷) → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
159 fveq2 6667 . . . . . 6 (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
160159breq2d 5075 . . . . 5 (𝑧 = 𝑉 → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘𝑉)))
161 oveq1 7155 . . . . . 6 (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
162161sseq1d 4002 . . . . 5 (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
163160, 162anbi12d 630 . . . 4 (𝑧 = 𝑉 → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
164163rspcev 3627 . . 3 ((𝑉 ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
165153, 157, 158, 164syl12anc 834 . 2 ((𝜑𝐸𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
166151, 165pm2.61dane 3109 1 (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144  {crab 3147  Vcvv 3500  cdif 3937  cun 3938  wss 3940  ifcif 4470  𝒫 cpw 4542  {csn 4564   class class class wbr 5063  cmpt 5143  ccnv 5553  cima 5557   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  Fincfn 8498  cc 10524  cr 10525  1c1 10527   + caddc 10529  cle 10665  cmin 10859  cn 11627  0cn0 11886  chash 13680   Ramsey cram 16325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681
This theorem is referenced by:  ramub1lem2  16353
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