MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ramub1lem1 Structured version   Visualization version   GIF version

Theorem ramub1lem1 16898
Description: Lemma for ramub1 16900. (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 3954 . . . . . . 7 (𝜑𝑉 ⊆ (𝑆 ∖ {𝑋}))
54difss2d 4094 . . . . . 6 (𝜑𝑉𝑆)
6 ramub1.x . . . . . . 7 (𝜑𝑋𝑆)
76snssd 4769 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝑆)
85, 7unssd 4146 . . . . 5 (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
91, 8sselpwd 5283 . . . 4 (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
109adantr 481 . . 3 ((𝜑𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
11 iftrue 4492 . . . . . . 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 5129 . . . . 5 ((𝜑𝐸 = 𝐷) → ((𝐹𝐷) − 1) ≤ (♯‘𝑉))
16 ramub1.f . . . . . . . . 9 (𝜑𝐹:𝑅⟶ℕ)
17 ramub1.d . . . . . . . . 9 (𝜑𝐷𝑅)
1816, 17ffvelcdmd 7036 . . . . . . . 8 (𝜑 → (𝐹𝐷) ∈ ℕ)
1918adantr 481 . . . . . . 7 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℕ)
2019nnred 12168 . . . . . 6 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℝ)
21 1red 11156 . . . . . 6 ((𝜑𝐸 = 𝐷) → 1 ∈ ℝ)
221, 5ssfid 9211 . . . . . . . 8 (𝜑𝑉 ∈ Fin)
23 hashcl 14256 . . . . . . . 8 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
24 nn0re 12422 . . . . . . . 8 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
2522, 23, 243syl 18 . . . . . . 7 (𝜑 → (♯‘𝑉) ∈ ℝ)
2625adantr 481 . . . . . 6 ((𝜑𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
2720, 21, 26lesubaddd 11752 . . . . 5 ((𝜑𝐸 = 𝐷) → (((𝐹𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
2815, 27mpbid 231 . . . 4 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ≤ ((♯‘𝑉) + 1))
29 fveq2 6842 . . . . 5 (𝐸 = 𝐷 → (𝐹𝐸) = (𝐹𝐷))
30 snidg 4620 . . . . . . . 8 (𝑋𝑆𝑋 ∈ {𝑋})
316, 30syl 17 . . . . . . 7 (𝜑𝑋 ∈ {𝑋})
324sseld 3943 . . . . . . . 8 (𝜑 → (𝑋𝑉𝑋 ∈ (𝑆 ∖ {𝑋})))
33 eldifn 4087 . . . . . . . 8 (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
3432, 33syl6 35 . . . . . . 7 (𝜑 → (𝑋𝑉 → ¬ 𝑋 ∈ {𝑋}))
3531, 34mt2d 136 . . . . . 6 (𝜑 → ¬ 𝑋𝑉)
36 hashunsng 14292 . . . . . . 7 (𝑋𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
376, 36syl 17 . . . . . 6 (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
3822, 35, 37mp2and 697 . . . . 5 (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
3929, 38breqan12rd 5122 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
4028, 39mpbird 256 . . 3 ((𝜑𝐸 = 𝐷) → (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})))
41 snfi 8988 . . . . . . 7 {𝑋} ∈ Fin
42 unfi 9116 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
4322, 41, 42sylancl 586 . . . . . 6 (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
44 ramub1.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
4544nnnn0d 12473 . . . . . 6 (𝜑𝑀 ∈ ℕ0)
46 ramub1.3 . . . . . . 7 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
4746hashbcval 16874 . . . . . 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 1224 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
5146hashbcval 16874 . . . . . . . . . 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 3983 . . . . . . . 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 1193 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
58 rabid 3427 . . . . . . . 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 3945 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
61 simpl2 1192 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
6261elpwid 4569 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
63 simpl1l 1224 . . . . . . . . . . . 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 3954 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥𝑆)
66 vex 3449 . . . . . . . . . . 11 𝑥 ∈ V
6766elpw 4564 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑆𝑥𝑆)
6865, 67sylibr 233 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
69 simpl3 1193 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
70 rabid 3427 . . . . . . . . 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 16874 . . . . . . . . . 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 2841 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
763difss2d 4094 . . . . . . . . . . . . . . 15 (𝜑𝑊𝑆)
771, 76ssfid 9211 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ Fin)
78 nnm1nn0 12454 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
7944, 78syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8046hashbcval 16874 . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . 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 6851 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
86 uncom 4113 . . . . . . . . . . . . . . . 16 (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
8762, 86sseqtrdi 3994 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
88 ssundif 4445 . . . . . . . . . . . . . . 15 (𝑥 ⊆ ({𝑋} ∪ 𝑉) ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑉)
8987, 88sylib 217 . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
9266difexi 5285 . . . . . . . . . . . . . 14 (𝑥 ∖ {𝑋}) ∈ V
9392elpw 4564 . . . . . . . . . . . . 13 ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑊)
9491, 93sylibr 233 . . . . . . . . . . . 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 9211 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
97 diffi 9123 . . . . . . . . . . . . . . . 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 14256 . . . . . . . . . . . . . . 15 ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
100 nn0cn 12423 . . . . . . . . . . . . . . 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 11109 . . . . . . . . . . . . . 14 1 ∈ ℂ
103 pncan 11407 . . . . . . . . . . . . . 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 4753 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
106 hashunsng 14292 . . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
109 undif1 4435 . . . . . . . . . . . . . . . . . 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 3921 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
112 elpwunsn 4644 . . . . . . . . . . . . . . . . . . . . 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 4769 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
115 ssequn2 4143 . . . . . . . . . . . . . . . . . . 19 ({𝑋} ⊆ 𝑥 ↔ (𝑥 ∪ {𝑋}) = 𝑥)
116114, 115sylib 217 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∪ {𝑋}) = 𝑥)
117109, 116eqtr2id 2789 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
118117fveq2d 6846 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
119118, 69eqtr3d 2778 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
120108, 119eqtr3d 2778 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
121120oveq1d 7372 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
122104, 121eqtr3d 2778 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
12385, 94, 122elrabd 3647 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
12484, 123sseldd 3945 . . . . . . . . . 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 6191 . . . . . . . . . . 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 2840 . . . . . . . . 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 4116 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
130129fveqeq2d 6850 . . . . . . . . . . 11 (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
131130elrab 3645 . . . . . . . . . 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 6846 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
135 simpl1r 1225 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
136133, 134, 1353eqtr4d 2786 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = 𝐸)
137 ramub1.6 . . . . . . . . 9 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
138137ffnd 6669 . . . . . . . 8 (𝜑𝐾 Fn (𝑆𝐶𝑀))
139 fniniseg 7010 . . . . . . . 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 711 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
14260, 141pm2.61dan 811 . . . . 5 (((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) → 𝑥 ∈ (𝐾 “ {𝐸}))
143142rabssdv 4032 . . . 4 ((𝜑𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
14449, 143eqsstrd 3982 . . 3 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
145 fveq2 6842 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
146145breq2d 5117 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
147 oveq1 7364 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
148147sseq1d 3975 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
149146, 148anbi12d 631 . . . 4 (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
150149rspcev 3581 . . 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 835 . 2 ((𝜑𝐸 = 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
1521, 5sselpwd 5283 . . . 4 (𝜑𝑉 ∈ 𝒫 𝑆)
153152adantr 481 . . 3 ((𝜑𝐸𝐷) → 𝑉 ∈ 𝒫 𝑆)
154 ifnefalse 4498 . . . . 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 5129 . . 3 ((𝜑𝐸𝐷) → (𝐹𝐸) ≤ (♯‘𝑉))
15853adantr 481 . . 3 ((𝜑𝐸𝐷) → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
159 fveq2 6842 . . . . . 6 (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
160159breq2d 5117 . . . . 5 (𝑧 = 𝑉 → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘𝑉)))
161 oveq1 7364 . . . . . 6 (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
162161sseq1d 3975 . . . . 5 (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
163160, 162anbi12d 631 . . . 4 (𝑧 = 𝑉 → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
164163rspcev 3581 . . 3 ((𝑉 ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
165153, 157, 158, 164syl12anc 835 . 2 ((𝜑𝐸𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
166151, 165pm2.61dane 3032 1 (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  cun 3908  wss 3910  ifcif 4486  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  cc 11049  cr 11050  1c1 11052   + caddc 11054  cle 11190  cmin 11385  cn 12153  0cn0 12413  chash 14230   Ramsey cram 16871
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-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-dju 9837  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-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231
This theorem is referenced by:  ramub1lem2  16899
  Copyright terms: Public domain W3C validator