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Theorem ram0 15707
Description: The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
ram0 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀)

Proof of Theorem ram0
Dummy variables 𝑏 𝑓 𝑐 𝑠 𝑥 𝑎 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2620 . . 3 (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖}) = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
2 id 22 . . 3 (𝑀 ∈ ℕ0𝑀 ∈ ℕ0)
3 0ex 4781 . . . 4 ∅ ∈ V
43a1i 11 . . 3 (𝑀 ∈ ℕ0 → ∅ ∈ V)
5 f0 6073 . . . 4 ∅:∅⟶ℕ0
65a1i 11 . . 3 (𝑀 ∈ ℕ0 → ∅:∅⟶ℕ0)
7 f00 6074 . . . . 5 (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶∅ ↔ (𝑓 = ∅ ∧ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅))
8 vex 3198 . . . . . . . . . 10 𝑠 ∈ V
9 simpl 473 . . . . . . . . . 10 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → 𝑀 ∈ ℕ0)
101hashbcval 15687 . . . . . . . . . 10 ((𝑠 ∈ V ∧ 𝑀 ∈ ℕ0) → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = {𝑥 ∈ 𝒫 𝑠 ∣ (#‘𝑥) = 𝑀})
118, 9, 10sylancr 694 . . . . . . . . 9 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = {𝑥 ∈ 𝒫 𝑠 ∣ (#‘𝑥) = 𝑀})
12 hashfz1 13117 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
1312breq1d 4654 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ0 → ((#‘(1...𝑀)) ≤ (#‘𝑠) ↔ 𝑀 ≤ (#‘𝑠)))
1413biimpar 502 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (#‘(1...𝑀)) ≤ (#‘𝑠))
15 fzfid 12755 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (1...𝑀) ∈ Fin)
16 hashdom 13151 . . . . . . . . . . . . . . 15 (((1...𝑀) ∈ Fin ∧ 𝑠 ∈ V) → ((#‘(1...𝑀)) ≤ (#‘𝑠) ↔ (1...𝑀) ≼ 𝑠))
1715, 8, 16sylancl 693 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ((#‘(1...𝑀)) ≤ (#‘𝑠) ↔ (1...𝑀) ≼ 𝑠))
1814, 17mpbid 222 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (1...𝑀) ≼ 𝑠)
198domen 7953 . . . . . . . . . . . . 13 ((1...𝑀) ≼ 𝑠 ↔ ∃𝑥((1...𝑀) ≈ 𝑥𝑥𝑠))
2018, 19sylib 208 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ∃𝑥((1...𝑀) ≈ 𝑥𝑥𝑠))
21 simprr 795 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → 𝑥𝑠)
22 selpw 4156 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 𝑠𝑥𝑠)
2321, 22sylibr 224 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → 𝑥 ∈ 𝒫 𝑠)
24 hasheni 13119 . . . . . . . . . . . . . . . . 17 ((1...𝑀) ≈ 𝑥 → (#‘(1...𝑀)) = (#‘𝑥))
2524ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → (#‘(1...𝑀)) = (#‘𝑥))
2612ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → (#‘(1...𝑀)) = 𝑀)
2725, 26eqtr3d 2656 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → (#‘𝑥) = 𝑀)
2823, 27jca 554 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → (𝑥 ∈ 𝒫 𝑠 ∧ (#‘𝑥) = 𝑀))
2928ex 450 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (((1...𝑀) ≈ 𝑥𝑥𝑠) → (𝑥 ∈ 𝒫 𝑠 ∧ (#‘𝑥) = 𝑀)))
3029eximdv 1844 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (∃𝑥((1...𝑀) ≈ 𝑥𝑥𝑠) → ∃𝑥(𝑥 ∈ 𝒫 𝑠 ∧ (#‘𝑥) = 𝑀)))
3120, 30mpd 15 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ∃𝑥(𝑥 ∈ 𝒫 𝑠 ∧ (#‘𝑥) = 𝑀))
32 df-rex 2915 . . . . . . . . . . 11 (∃𝑥 ∈ 𝒫 𝑠(#‘𝑥) = 𝑀 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝑠 ∧ (#‘𝑥) = 𝑀))
3331, 32sylibr 224 . . . . . . . . . 10 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ∃𝑥 ∈ 𝒫 𝑠(#‘𝑥) = 𝑀)
34 rabn0 3949 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 𝑠 ∣ (#‘𝑥) = 𝑀} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝑠(#‘𝑥) = 𝑀)
3533, 34sylibr 224 . . . . . . . . 9 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → {𝑥 ∈ 𝒫 𝑠 ∣ (#‘𝑥) = 𝑀} ≠ ∅)
3611, 35eqnetrd 2858 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ≠ ∅)
3736neneqd 2796 . . . . . . 7 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ¬ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅)
3837pm2.21d 118 . . . . . 6 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ((𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅ → ∃𝑐 ∈ ∅ ∃𝑥 ∈ 𝒫 𝑠((∅‘𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐}))))
3938adantld 483 . . . . 5 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → ((𝑓 = ∅ ∧ (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅) → ∃𝑐 ∈ ∅ ∃𝑥 ∈ 𝒫 𝑠((∅‘𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐}))))
407, 39syl5bi 232 . . . 4 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶∅ → ∃𝑐 ∈ ∅ ∃𝑥 ∈ 𝒫 𝑠((∅‘𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐}))))
4140impr 648 . . 3 ((𝑀 ∈ ℕ0 ∧ (𝑀 ≤ (#‘𝑠) ∧ 𝑓:(𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶∅)) → ∃𝑐 ∈ ∅ ∃𝑥 ∈ 𝒫 𝑠((∅‘𝑐) ≤ (#‘𝑥) ∧ (𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (𝑓 “ {𝑐})))
421, 2, 4, 6, 2, 41ramub 15698 . 2 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ≤ 𝑀)
43 nnnn0 11284 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
443a1i 11 . . . . . 6 (𝑀 ∈ ℕ → ∅ ∈ V)
455a1i 11 . . . . . 6 (𝑀 ∈ ℕ → ∅:∅⟶ℕ0)
46 nnm1nn0 11319 . . . . . 6 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
47 f0 6073 . . . . . . 7 ∅:∅⟶∅
48 fzfid 12755 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (1...(𝑀 − 1)) ∈ Fin)
491hashbc2 15691 . . . . . . . . . . 11 (((1...(𝑀 − 1)) ∈ Fin ∧ 𝑀 ∈ ℕ0) → (#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = ((#‘(1...(𝑀 − 1)))C𝑀))
5048, 43, 49syl2anc 692 . . . . . . . . . 10 (𝑀 ∈ ℕ → (#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = ((#‘(1...(𝑀 − 1)))C𝑀))
51 hashfz1 13117 . . . . . . . . . . . 12 ((𝑀 − 1) ∈ ℕ0 → (#‘(1...(𝑀 − 1))) = (𝑀 − 1))
5246, 51syl 17 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (#‘(1...(𝑀 − 1))) = (𝑀 − 1))
5352oveq1d 6650 . . . . . . . . . 10 (𝑀 ∈ ℕ → ((#‘(1...(𝑀 − 1)))C𝑀) = ((𝑀 − 1)C𝑀))
54 nnz 11384 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
55 nnre 11012 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5655ltm1d 10941 . . . . . . . . . . . 12 (𝑀 ∈ ℕ → (𝑀 − 1) < 𝑀)
5756olcd 408 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (𝑀 < 0 ∨ (𝑀 − 1) < 𝑀))
58 bcval4 13077 . . . . . . . . . . 11 (((𝑀 − 1) ∈ ℕ0𝑀 ∈ ℤ ∧ (𝑀 < 0 ∨ (𝑀 − 1) < 𝑀)) → ((𝑀 − 1)C𝑀) = 0)
5946, 54, 57, 58syl3anc 1324 . . . . . . . . . 10 (𝑀 ∈ ℕ → ((𝑀 − 1)C𝑀) = 0)
6050, 53, 593eqtrd 2658 . . . . . . . . 9 (𝑀 ∈ ℕ → (#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = 0)
61 ovex 6663 . . . . . . . . . 10 ((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ∈ V
62 hasheq0 13137 . . . . . . . . . 10 (((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ∈ V → ((#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = 0 ↔ ((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅))
6361, 62ax-mp 5 . . . . . . . . 9 ((#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = 0 ↔ ((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅)
6460, 63sylib 208 . . . . . . . 8 (𝑀 ∈ ℕ → ((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) = ∅)
6564feq2d 6018 . . . . . . 7 (𝑀 ∈ ℕ → (∅:((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶∅ ↔ ∅:∅⟶∅))
6647, 65mpbiri 248 . . . . . 6 (𝑀 ∈ ℕ → ∅:((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)⟶∅)
67 noel 3911 . . . . . . . 8 ¬ 𝑐 ∈ ∅
6867pm2.21i 116 . . . . . . 7 (𝑐 ∈ ∅ → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (∅ “ {𝑐}) → (#‘𝑥) < (∅‘𝑐)))
6968ad2antrl 763 . . . . . 6 ((𝑀 ∈ ℕ ∧ (𝑐 ∈ ∅ ∧ 𝑥 ⊆ (1...(𝑀 − 1)))) → ((𝑥(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ⊆ (∅ “ {𝑐}) → (#‘𝑥) < (∅‘𝑐)))
701, 43, 44, 45, 46, 66, 69ramlb 15704 . . . . 5 (𝑀 ∈ ℕ → (𝑀 − 1) < (𝑀 Ramsey ∅))
71 ramubcl 15703 . . . . . . . 8 (((𝑀 ∈ ℕ0 ∧ ∅ ∈ V ∧ ∅:∅⟶ℕ0) ∧ (𝑀 ∈ ℕ0 ∧ (𝑀 Ramsey ∅) ≤ 𝑀)) → (𝑀 Ramsey ∅) ∈ ℕ0)
722, 4, 6, 2, 42, 71syl32anc 1332 . . . . . . 7 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ∈ ℕ0)
7343, 72syl 17 . . . . . 6 (𝑀 ∈ ℕ → (𝑀 Ramsey ∅) ∈ ℕ0)
74 nn0lem1lt 11427 . . . . . 6 ((𝑀 ∈ ℕ0 ∧ (𝑀 Ramsey ∅) ∈ ℕ0) → (𝑀 ≤ (𝑀 Ramsey ∅) ↔ (𝑀 − 1) < (𝑀 Ramsey ∅)))
7543, 73, 74syl2anc 692 . . . . 5 (𝑀 ∈ ℕ → (𝑀 ≤ (𝑀 Ramsey ∅) ↔ (𝑀 − 1) < (𝑀 Ramsey ∅)))
7670, 75mpbird 247 . . . 4 (𝑀 ∈ ℕ → 𝑀 ≤ (𝑀 Ramsey ∅))
7776a1i 11 . . 3 (𝑀 ∈ ℕ0 → (𝑀 ∈ ℕ → 𝑀 ≤ (𝑀 Ramsey ∅)))
7872nn0ge0d 11339 . . . 4 (𝑀 ∈ ℕ0 → 0 ≤ (𝑀 Ramsey ∅))
79 breq1 4647 . . . 4 (𝑀 = 0 → (𝑀 ≤ (𝑀 Ramsey ∅) ↔ 0 ≤ (𝑀 Ramsey ∅)))
8078, 79syl5ibrcom 237 . . 3 (𝑀 ∈ ℕ0 → (𝑀 = 0 → 𝑀 ≤ (𝑀 Ramsey ∅)))
81 elnn0 11279 . . . 4 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
8281biimpi 206 . . 3 (𝑀 ∈ ℕ0 → (𝑀 ∈ ℕ ∨ 𝑀 = 0))
8377, 80, 82mpjaod 396 . 2 (𝑀 ∈ ℕ0𝑀 ≤ (𝑀 Ramsey ∅))
8472nn0red 11337 . . 3 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ∈ ℝ)
85 nn0re 11286 . . 3 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
8684, 85letri3d 10164 . 2 (𝑀 ∈ ℕ0 → ((𝑀 Ramsey ∅) = 𝑀 ↔ ((𝑀 Ramsey ∅) ≤ 𝑀𝑀 ≤ (𝑀 Ramsey ∅))))
8742, 83, 86mpbir2and 956 1 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  wrex 2910  {crab 2913  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149  {csn 4168   class class class wbr 4644  ccnv 5103  cima 5107  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  cen 7937  cdom 7938  Fincfn 7940  0cc0 9921  1c1 9922   < clt 10059  cle 10060  cmin 10251  cn 11005  0cn0 11277  cz 11362  ...cfz 12311  Ccbc 13072  #chash 13100   Ramsey cram 15684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-seq 12785  df-fac 13044  df-bc 13073  df-hash 13101  df-ram 15686
This theorem is referenced by:  0ramcl  15708  ramcl  15714
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