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Theorem erdszelem10 34680
Description: Lemma for erdsze 34682. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (πœ‘ β†’ 𝑁 ∈ β„•)
erdsze.f (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i 𝐼 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
erdszelem.j 𝐽 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
erdszelem.t 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)
erdszelem.r (πœ‘ β†’ 𝑅 ∈ β„•)
erdszelem.s (πœ‘ β†’ 𝑆 ∈ β„•)
erdszelem.m (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < 𝑁)
Assertion
Ref Expression
erdszelem10 (πœ‘ β†’ βˆƒπ‘š ∈ (1...𝑁)(Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))))
Distinct variable groups:   π‘₯,𝑦   π‘š,𝑛,π‘₯,𝑦,𝐹   𝑛,𝐼,π‘₯,𝑦   𝑛,𝐽,π‘₯,𝑦   𝑅,π‘š,π‘₯,𝑦   π‘š,𝑁,𝑛,π‘₯,𝑦   πœ‘,π‘š,𝑛,π‘₯,𝑦   𝑆,π‘š,π‘₯,𝑦   𝑇,π‘š
Allowed substitution hints:   𝑅(𝑛)   𝑆(𝑛)   𝑇(π‘₯,𝑦,𝑛)   𝐼(π‘š)   𝐽(π‘š)

Proof of Theorem erdszelem10
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 fzfi 13934 . . . . . . . 8 (1...(𝑅 βˆ’ 1)) ∈ Fin
2 fzfi 13934 . . . . . . . 8 (1...(𝑆 βˆ’ 1)) ∈ Fin
3 xpfi 9313 . . . . . . . 8 (((1...(𝑅 βˆ’ 1)) ∈ Fin ∧ (1...(𝑆 βˆ’ 1)) ∈ Fin) β†’ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ∈ Fin)
41, 2, 3mp2an 689 . . . . . . 7 ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ∈ Fin
5 ssdomg 8992 . . . . . . 7 (((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ∈ Fin β†’ (ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β†’ ran 𝑇 β‰Ό ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
64, 5ax-mp 5 . . . . . 6 (ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β†’ ran 𝑇 β‰Ό ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))))
7 domnsym 9095 . . . . . 6 (ran 𝑇 β‰Ό ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β†’ Β¬ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί ran 𝑇)
86, 7syl 17 . . . . 5 (ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β†’ Β¬ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί ran 𝑇)
9 erdszelem.m . . . . . . . 8 (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < 𝑁)
10 hashxp 14391 . . . . . . . . . 10 (((1...(𝑅 βˆ’ 1)) ∈ Fin ∧ (1...(𝑆 βˆ’ 1)) ∈ Fin) β†’ (β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) = ((β™―β€˜(1...(𝑅 βˆ’ 1))) Β· (β™―β€˜(1...(𝑆 βˆ’ 1)))))
111, 2, 10mp2an 689 . . . . . . . . 9 (β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) = ((β™―β€˜(1...(𝑅 βˆ’ 1))) Β· (β™―β€˜(1...(𝑆 βˆ’ 1))))
12 erdszelem.r . . . . . . . . . . 11 (πœ‘ β†’ 𝑅 ∈ β„•)
13 nnm1nn0 12510 . . . . . . . . . . 11 (𝑅 ∈ β„• β†’ (𝑅 βˆ’ 1) ∈ β„•0)
14 hashfz1 14303 . . . . . . . . . . 11 ((𝑅 βˆ’ 1) ∈ β„•0 β†’ (β™―β€˜(1...(𝑅 βˆ’ 1))) = (𝑅 βˆ’ 1))
1512, 13, 143syl 18 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜(1...(𝑅 βˆ’ 1))) = (𝑅 βˆ’ 1))
16 erdszelem.s . . . . . . . . . . 11 (πœ‘ β†’ 𝑆 ∈ β„•)
17 nnm1nn0 12510 . . . . . . . . . . 11 (𝑆 ∈ β„• β†’ (𝑆 βˆ’ 1) ∈ β„•0)
18 hashfz1 14303 . . . . . . . . . . 11 ((𝑆 βˆ’ 1) ∈ β„•0 β†’ (β™―β€˜(1...(𝑆 βˆ’ 1))) = (𝑆 βˆ’ 1))
1916, 17, 183syl 18 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜(1...(𝑆 βˆ’ 1))) = (𝑆 βˆ’ 1))
2015, 19oveq12d 7419 . . . . . . . . 9 (πœ‘ β†’ ((β™―β€˜(1...(𝑅 βˆ’ 1))) Β· (β™―β€˜(1...(𝑆 βˆ’ 1)))) = ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)))
2111, 20eqtrid 2776 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) = ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)))
22 erdsze.n . . . . . . . . . 10 (πœ‘ β†’ 𝑁 ∈ β„•)
2322nnnn0d 12529 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ β„•0)
24 hashfz1 14303 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (β™―β€˜(1...𝑁)) = 𝑁)
2523, 24syl 17 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜(1...𝑁)) = 𝑁)
269, 21, 253brtr4d 5170 . . . . . . 7 (πœ‘ β†’ (β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) < (β™―β€˜(1...𝑁)))
27 fzfid 13935 . . . . . . . 8 (πœ‘ β†’ (1...𝑁) ∈ Fin)
28 hashsdom 14338 . . . . . . . 8 ((((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ ((β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) < (β™―β€˜(1...𝑁)) ↔ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί (1...𝑁)))
294, 27, 28sylancr 586 . . . . . . 7 (πœ‘ β†’ ((β™―β€˜((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))) < (β™―β€˜(1...𝑁)) ↔ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί (1...𝑁)))
3026, 29mpbid 231 . . . . . 6 (πœ‘ β†’ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί (1...𝑁))
31 erdsze.f . . . . . . . 8 (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)
32 erdszelem.i . . . . . . . 8 𝐼 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
33 erdszelem.j . . . . . . . 8 𝐽 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
34 erdszelem.t . . . . . . . 8 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)
3522, 31, 32, 33, 34erdszelem9 34679 . . . . . . 7 (πœ‘ β†’ 𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•))
36 f1f1orn 6834 . . . . . . 7 (𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•) β†’ 𝑇:(1...𝑁)–1-1-ontoβ†’ran 𝑇)
37 ovex 7434 . . . . . . . 8 (1...𝑁) ∈ V
3837f1oen 8965 . . . . . . 7 (𝑇:(1...𝑁)–1-1-ontoβ†’ran 𝑇 β†’ (1...𝑁) β‰ˆ ran 𝑇)
3935, 36, 383syl 18 . . . . . 6 (πœ‘ β†’ (1...𝑁) β‰ˆ ran 𝑇)
40 sdomentr 9107 . . . . . 6 ((((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί (1...𝑁) ∧ (1...𝑁) β‰ˆ ran 𝑇) β†’ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί ran 𝑇)
4130, 39, 40syl2anc 583 . . . . 5 (πœ‘ β†’ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) β‰Ί ran 𝑇)
428, 41nsyl3 138 . . . 4 (πœ‘ β†’ Β¬ ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))))
43 nss 4038 . . . . 5 (Β¬ ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘ (𝑠 ∈ ran 𝑇 ∧ Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
44 df-rex 3063 . . . . 5 (βˆƒπ‘  ∈ ran 𝑇 Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘ (𝑠 ∈ ran 𝑇 ∧ Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
4543, 44bitr4i 278 . . . 4 (Β¬ ran 𝑇 βŠ† ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘  ∈ ran 𝑇 Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))))
4642, 45sylib 217 . . 3 (πœ‘ β†’ βˆƒπ‘  ∈ ran 𝑇 Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))))
47 f1fn 6778 . . . 4 (𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•) β†’ 𝑇 Fn (1...𝑁))
48 eleq1 2813 . . . . . 6 (𝑠 = (π‘‡β€˜π‘š) β†’ (𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
4948notbid 318 . . . . 5 (𝑠 = (π‘‡β€˜π‘š) β†’ (Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
5049rexrn 7078 . . . 4 (𝑇 Fn (1...𝑁) β†’ (βˆƒπ‘  ∈ ran 𝑇 Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘š ∈ (1...𝑁) Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
5135, 47, 503syl 18 . . 3 (πœ‘ β†’ (βˆƒπ‘  ∈ ran 𝑇 Β¬ 𝑠 ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘š ∈ (1...𝑁) Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
5246, 51mpbid 231 . 2 (πœ‘ β†’ βˆƒπ‘š ∈ (1...𝑁) Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))))
53 fveq2 6881 . . . . . . . . . 10 (𝑛 = π‘š β†’ (πΌβ€˜π‘›) = (πΌβ€˜π‘š))
54 fveq2 6881 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π½β€˜π‘›) = (π½β€˜π‘š))
5553, 54opeq12d 4873 . . . . . . . . 9 (𝑛 = π‘š β†’ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩ = ⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩)
56 opex 5454 . . . . . . . . 9 ⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩ ∈ V
5755, 34, 56fvmpt 6988 . . . . . . . 8 (π‘š ∈ (1...𝑁) β†’ (π‘‡β€˜π‘š) = ⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩)
5857adantl 481 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (1...𝑁)) β†’ (π‘‡β€˜π‘š) = ⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩)
5958eleq1d 2810 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (1...𝑁)) β†’ ((π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ ⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩ ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1)))))
60 opelxp 5702 . . . . . 6 (⟨(πΌβ€˜π‘š), (π½β€˜π‘š)⟩ ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ ((πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∧ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))))
6159, 60bitrdi 287 . . . . 5 ((πœ‘ ∧ π‘š ∈ (1...𝑁)) β†’ ((π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ ((πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∧ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1)))))
6261notbid 318 . . . 4 ((πœ‘ ∧ π‘š ∈ (1...𝑁)) β†’ (Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ Β¬ ((πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∧ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1)))))
63 ianor 978 . . . 4 (Β¬ ((πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∧ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))) ↔ (Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))))
6462, 63bitrdi 287 . . 3 ((πœ‘ ∧ π‘š ∈ (1...𝑁)) β†’ (Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ (Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1)))))
6564rexbidva 3168 . 2 (πœ‘ β†’ (βˆƒπ‘š ∈ (1...𝑁) Β¬ (π‘‡β€˜π‘š) ∈ ((1...(𝑅 βˆ’ 1)) Γ— (1...(𝑆 βˆ’ 1))) ↔ βˆƒπ‘š ∈ (1...𝑁)(Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1)))))
6652, 65mpbid 231 1 (πœ‘ β†’ βˆƒπ‘š ∈ (1...𝑁)(Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424   βŠ† wss 3940  π’« cpw 4594  βŸ¨cop 4626   class class class wbr 5138   ↦ cmpt 5221   Γ— cxp 5664  β—‘ccnv 5665  ran crn 5667   β†Ύ cres 5668   β€œ cima 5669   Fn wfn 6528  β€“1-1β†’wf1 6530  β€“1-1-ontoβ†’wf1o 6532  β€˜cfv 6533   Isom wiso 6534  (class class class)co 7401   β‰ˆ cen 8932   β‰Ό cdom 8933   β‰Ί csdm 8934  Fincfn 8935  supcsup 9431  β„cr 11105  1c1 11107   Β· cmul 11111   < clt 11245   βˆ’ cmin 11441  β„•cn 12209  β„•0cn0 12469  ...cfz 13481  β™―chash 14287
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-fz 13482  df-hash 14288
This theorem is referenced by:  erdszelem11  34681
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