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Theorem ram0 15645
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 2626 . . 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 4755 . . . 4 ∅ ∈ V
43a1i 11 . . 3 (𝑀 ∈ ℕ0 → ∅ ∈ V)
5 f0 6045 . . . 4 ∅:∅⟶ℕ0
65a1i 11 . . 3 (𝑀 ∈ ℕ0 → ∅:∅⟶ℕ0)
7 f00 6046 . . . . 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 3194 . . . . . . . . . 10 𝑠 ∈ V
9 simpl 473 . . . . . . . . . 10 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → 𝑀 ∈ ℕ0)
101hashbcval 15625 . . . . . . . . . 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 13071 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
1312breq1d 4628 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ0 → ((#‘(1...𝑀)) ≤ (#‘𝑠) ↔ 𝑀 ≤ (#‘𝑠)))
1413biimpar 502 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (#‘(1...𝑀)) ≤ (#‘𝑠))
15 fzfid 12709 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (1...𝑀) ∈ Fin)
16 hashdom 13105 . . . . . . . . . . . . . . 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 7913 . . . . . . . . . . . . 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 4142 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 𝑠𝑥𝑠)
2321, 22sylibr 224 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) ∧ ((1...𝑀) ≈ 𝑥𝑥𝑠)) → 𝑥 ∈ 𝒫 𝑠)
24 hasheni 13073 . . . . . . . . . . . . . . . . 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 2662 . . . . . . . . . . . . . . 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 1848 . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 3937 . . . . . . . . . 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 2863 . . . . . . . 8 ((𝑀 ∈ ℕ0𝑀 ≤ (#‘𝑠)) → (𝑠(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ≠ ∅)
3736neneqd 2801 . . . . . . 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 15636 . 2 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ≤ 𝑀)
43 nnnn0 11244 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
443a1i 11 . . . . . 6 (𝑀 ∈ ℕ → ∅ ∈ V)
455a1i 11 . . . . . 6 (𝑀 ∈ ℕ → ∅:∅⟶ℕ0)
46 nnm1nn0 11279 . . . . . 6 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
47 f0 6045 . . . . . . 7 ∅:∅⟶∅
48 fzfid 12709 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (1...(𝑀 − 1)) ∈ Fin)
491hashbc2 15629 . . . . . . . . . . 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 13071 . . . . . . . . . . . 12 ((𝑀 − 1) ∈ ℕ0 → (#‘(1...(𝑀 − 1))) = (𝑀 − 1))
5246, 51syl 17 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (#‘(1...(𝑀 − 1))) = (𝑀 − 1))
5352oveq1d 6620 . . . . . . . . . 10 (𝑀 ∈ ℕ → ((#‘(1...(𝑀 − 1)))C𝑀) = ((𝑀 − 1)C𝑀))
54 nnz 11344 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
55 nnre 10972 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5655ltm1d 10901 . . . . . . . . . . . 12 (𝑀 ∈ ℕ → (𝑀 − 1) < 𝑀)
5756olcd 408 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (𝑀 < 0 ∨ (𝑀 − 1) < 𝑀))
58 bcval4 13031 . . . . . . . . . . 11 (((𝑀 − 1) ∈ ℕ0𝑀 ∈ ℤ ∧ (𝑀 < 0 ∨ (𝑀 − 1) < 𝑀)) → ((𝑀 − 1)C𝑀) = 0)
5946, 54, 57, 58syl3anc 1323 . . . . . . . . . 10 (𝑀 ∈ ℕ → ((𝑀 − 1)C𝑀) = 0)
6050, 53, 593eqtrd 2664 . . . . . . . . 9 (𝑀 ∈ ℕ → (#‘((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀)) = 0)
61 ovex 6633 . . . . . . . . . 10 ((1...(𝑀 − 1))(𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})𝑀) ∈ V
62 hasheq0 13091 . . . . . . . . . 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 5990 . . . . . . 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 3900 . . . . . . . 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 15642 . . . . 5 (𝑀 ∈ ℕ → (𝑀 − 1) < (𝑀 Ramsey ∅))
71 ramubcl 15641 . . . . . . . 8 (((𝑀 ∈ ℕ0 ∧ ∅ ∈ V ∧ ∅:∅⟶ℕ0) ∧ (𝑀 ∈ ℕ0 ∧ (𝑀 Ramsey ∅) ≤ 𝑀)) → (𝑀 Ramsey ∅) ∈ ℕ0)
722, 4, 6, 2, 42, 71syl32anc 1331 . . . . . . 7 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ∈ ℕ0)
7343, 72syl 17 . . . . . 6 (𝑀 ∈ ℕ → (𝑀 Ramsey ∅) ∈ ℕ0)
74 nn0lem1lt 11386 . . . . . 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 11299 . . . 4 (𝑀 ∈ ℕ0 → 0 ≤ (𝑀 Ramsey ∅))
79 breq1 4621 . . . 4 (𝑀 = 0 → (𝑀 ≤ (𝑀 Ramsey ∅) ↔ 0 ≤ (𝑀 Ramsey ∅)))
8078, 79syl5ibrcom 237 . . 3 (𝑀 ∈ ℕ0 → (𝑀 = 0 → 𝑀 ≤ (𝑀 Ramsey ∅)))
81 elnn0 11239 . . . 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 11297 . . 3 (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) ∈ ℝ)
85 nn0re 11246 . . 3 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
8684, 85letri3d 10124 . 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 1480  wex 1701  wcel 1992  wne 2796  wrex 2913  {crab 2916  Vcvv 3191  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  ccnv 5078  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  cen 7897  cdom 7898  Fincfn 7900  0cc0 9881  1c1 9882   < clt 10019  cle 10020  cmin 10211  cn 10965  0cn0 11237  cz 11322  ...cfz 12265  Ccbc 13026  #chash 13054   Ramsey cram 15622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-seq 12739  df-fac 12998  df-bc 13027  df-hash 13055  df-ram 15624
This theorem is referenced by:  0ramcl  15646  ramcl  15652
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