erdsze2lem1 - Mathbox for Mario Carneiro

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Theorem erdsze2lem1 36134

Description: Lemma for erdsze2 36136. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze2.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdsze2.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdsze2.f	⊢ (𝜑 → 𝐹:𝐴⟶1-1ℝ)
erdsze2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
erdsze2lem.n	⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
erdsze2lem.l	⊢ (𝜑 → 𝑁 < (♯‘𝐴))

**Assertion**
Ref	Expression
erdsze2lem1	⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Distinct variable groups: 𝐴,𝑓 𝑓,𝐹 𝑅,𝑓 𝑆,𝑓 𝑓,𝑁 𝜑,𝑓

Proof of Theorem erdsze2lem1 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze2lem.n	. . . . . . . . 9 ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1))
2		erdsze2.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3		nnm1nn0 12671	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
4	2, 3	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 − 1) ∈ ℕ₀)
5		erdsze2.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ)
6		nnm1nn0 12671	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
7	5, 6	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 − 1) ∈ ℕ₀)
8	4, 7	nn0mulcld 12696	. . . . . . . . 9 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ₀)
9	1, 8	eqeltrid 2868	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		peano2nn0 12670	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
11		hashfz1 14514	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
12	9, 10, 11	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
13	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
14		erdsze2lem.l	. . . . . . . 8 ⊢ (𝜑 → 𝑁 < (♯‘𝐴))
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝑁 < (♯‘𝐴))
16		hashcl 14524	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
17		nn0ltp1le 12781	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
18	9, 16, 17	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
19	15, 18	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 + 1) ≤ (♯‘𝐴))
20	13, 19	eqbrtrd 5130	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴))
21		fzfid 14140	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
22		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
23		hashdom 14547	. . . . . 6 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
24	21, 22, 23	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
25	20, 24	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
26		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
27		fzfid 14140	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
28		isinffi 10097	. . . . . 6 ⊢ ((¬ 𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))⟶1-1𝐴)
29	26, 27, 28	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))⟶1-1𝐴)
30		erdsze2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
31		reex 11315	. . . . . . . 8 ⊢ ℝ ∈ V
32		ssexg 5285	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
33	30, 31, 32	sylancl 598	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
34	33	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ∈ V)
35		brdomg 9010	. . . . . 6 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))⟶1-1𝐴))
36	34, 35	syl 18	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))⟶1-1𝐴))
37	29, 36	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
38	25, 37	pm2.61dan 825	. . 3 ⊢ (𝜑 → (1...(𝑁 + 1)) ≼ 𝐴)
39		domeng 9014	. . . 4 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
40	33, 39	syl 18	. . 3 ⊢ (𝜑 → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
41	38, 40	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴))
42		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ 𝐴)
43	30	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝐴 ⊆ ℝ)
44	42, 43	sstrd 3923	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ ℝ)
45		ltso 11414	. . . . 5 ⊢ < Or ℝ
46		soss 5583	. . . . 5 ⊢ (𝑠 ⊆ ℝ → ( < Or ℝ → < Or 𝑠))
47	44, 45, 46	mpisyl 22	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → < Or 𝑠)
48		fzfid 14140	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ∈ Fin)
49		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ≈ 𝑠)
50		enfi 9227	. . . . . 6 ⊢ ((1...(𝑁 + 1)) ≈ 𝑠 → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
51	49, 50	syl 18	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
52	48, 51	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ∈ Fin)
53		fz1iso 14631	. . . 4 ⊢ (( < Or 𝑠 ∧ 𝑠 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
54	47, 52, 53	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
55		isof1o 7351	. . . . . . . . . 10 ⊢ (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → 𝑓:(1...(♯‘𝑠))⟶1-1onto𝑠)
56	55	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(♯‘𝑠))⟶1-1onto𝑠)
57		hashen 14515	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑠 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) = (♯‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
58	48, 52, 57	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((♯‘(1...(𝑁 + 1))) = (♯‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
59	49, 58	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘(1...(𝑁 + 1))) = (♯‘𝑠))
60	12	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
61	59, 60	eqtr3d 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘𝑠) = (𝑁 + 1))
62	61	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (♯‘𝑠) = (𝑁 + 1))
63	62	oveq2d 7456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (1...(♯‘𝑠)) = (1...(𝑁 + 1)))
64	63	f1oeq2d 6839	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓:(1...(♯‘𝑠))⟶1-1onto𝑠 ↔ 𝑓:(1...(𝑁 + 1))⟶1-1onto𝑠))
65	56, 64	mpbid 235	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))⟶1-1onto𝑠)
66		f1of1 6842	. . . . . . . 8 ⊢ (𝑓:(1...(𝑁 + 1))⟶1-1onto𝑠 → 𝑓:(1...(𝑁 + 1))⟶1-1𝑠)
67	65, 66	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))⟶1-1𝑠)
68		simplrr 790	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑠 ⊆ 𝐴)
69		f1ss 6804	. . . . . . 7 ⊢ ((𝑓:(1...(𝑁 + 1))⟶1-1𝑠 ∧ 𝑠 ⊆ 𝐴) → 𝑓:(1...(𝑁 + 1))⟶1-1𝐴)
70	67, 68, 69	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))⟶1-1𝐴)
71		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
72		f1ofo 6851	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝑠))⟶1-1onto𝑠 → 𝑓:(1...(♯‘𝑠))⟶onto𝑠)
73		forn 6818	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝑠))⟶onto𝑠 → ran 𝑓 = 𝑠)
74		isoeq5 7349	. . . . . . . . 9 ⊢ (ran 𝑓 = 𝑠 → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)))
75	56, 72, 73, 74	4syl 20	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)))
76	71, 75	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓))
77		isoeq4 7348	. . . . . . . 8 ⊢ ((1...(♯‘𝑠)) = (1...(𝑁 + 1)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
78	63, 77	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
79	76, 78	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))
80	70, 79	jca 521	. . . . 5 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
81	80	ex 418	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → (𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
82	81	eximdv 1952	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → ∃𝑓(𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
83	54, 82	mpd 16	. 2 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓(𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
84	41, 83	exlimddv 1970	1 ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))⟶1-1𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 class class class wbr 5106 Or wor 5562 ran crn 5656 ⟶1-1wf1 6552 ⟶ontowfo 6553 ⟶1-1ontowf1o 6554 ‘cfv 6555 Isom wiso 6556 (class class class)co 7440 ≈ cen 8995 ≼ cdom 8996 Fincfn 8998 ℝcr 11223 1c1 11225 + caddc 11227 · cmul 11229 < clt 11367 ≤ cle 11368 − cmin 11565 ℕcn 12359 ℕ₀cn0 12630 ...cfz 13663 ♯chash 14498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-oadd 8505 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-n0 12631 df-xnn0 12704 df-z 12718 df-uz 12990 df-fz 13664 df-hash 14499

This theorem is used by: erdsze2 36136

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