erdsze2lem1 - Mathbox for Mario Carneiro

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Theorem erdsze2lem1 34492

Description: Lemma for erdsze2 34494. (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–1-1→ℝ)
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))–1-1→𝐴 ∧ 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 12517	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈ ℕ₀)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 − 1) ∈ ℕ₀)
5		erdsze2.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ)
6		nnm1nn0 12517	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈ ℕ₀)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 − 1) ∈ ℕ₀)
8	4, 7	nn0mulcld 12541	. . . . . . . . 9 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈ ℕ₀)
9	1, 8	eqeltrid 2835	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		peano2nn0 12516	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
11		hashfz1 14310	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
12	9, 10, 11	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
13	12	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
14		erdsze2lem.l	. . . . . . . 8 ⊢ (𝜑 → 𝑁 < (♯‘𝐴))
15	14	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝑁 < (♯‘𝐴))
16		hashcl 14320	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
17		nn0ltp1le 12624	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
18	9, 16, 17	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) ≤ (♯‘𝐴)))
19	15, 18	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (𝑁 + 1) ≤ (♯‘𝐴))
20	13, 19	eqbrtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴))
21		fzfid 13942	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
22		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
23		hashdom 14343	. . . . . 6 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
24	21, 22, 23	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) ≤ (♯‘𝐴) ↔ (1...(𝑁 + 1)) ≼ 𝐴))
25	20, 24	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
26		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
27		fzfid 13942	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ∈ Fin)
28		isinffi 9989	. . . . . 6 ⊢ ((¬ 𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
29	26, 27, 28	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
30		erdsze2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
31		reex 11203	. . . . . . . 8 ⊢ ℝ ∈ V
32		ssexg 5322	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ∈ V) → 𝐴 ∈ V)
33	30, 31, 32	sylancl 584	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
34	33	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ∈ V)
35		brdomg 8954	. . . . . 6 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴))
36	34, 35	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑓 𝑓:(1...(𝑁 + 1))–1-1→𝐴))
37	29, 36	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ Fin) → (1...(𝑁 + 1)) ≼ 𝐴)
38	25, 37	pm2.61dan 809	. . 3 ⊢ (𝜑 → (1...(𝑁 + 1)) ≼ 𝐴)
39		domeng 8960	. . . 4 ⊢ (𝐴 ∈ V → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
40	33, 39	syl 17	. . 3 ⊢ (𝜑 → ((1...(𝑁 + 1)) ≼ 𝐴 ↔ ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)))
41	38, 40	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑠((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴))
42		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ 𝐴)
43	30	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝐴 ⊆ ℝ)
44	42, 43	sstrd 3991	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ⊆ ℝ)
45		ltso 11298	. . . . 5 ⊢ < Or ℝ
46		soss 5607	. . . . 5 ⊢ (𝑠 ⊆ ℝ → ( < Or ℝ → < Or 𝑠))
47	44, 45, 46	mpisyl 21	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → < Or 𝑠)
48		fzfid 13942	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ∈ Fin)
49		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (1...(𝑁 + 1)) ≈ 𝑠)
50		enfi 9192	. . . . . 6 ⊢ ((1...(𝑁 + 1)) ≈ 𝑠 → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
51	49, 50	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((1...(𝑁 + 1)) ∈ Fin ↔ 𝑠 ∈ Fin))
52	48, 51	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → 𝑠 ∈ Fin)
53		fz1iso 14427	. . . 4 ⊢ (( < Or 𝑠 ∧ 𝑠 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
54	47, 52, 53	syl2anc 582	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
55		isof1o 7322	. . . . . . . . . 10 ⊢ (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → 𝑓:(1...(♯‘𝑠))–1-1-onto→𝑠)
56	55	adantl 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(♯‘𝑠))–1-1-onto→𝑠)
57		hashen 14311	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑁 + 1)) ∈ Fin ∧ 𝑠 ∈ Fin) → ((♯‘(1...(𝑁 + 1))) = (♯‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
58	48, 52, 57	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ((♯‘(1...(𝑁 + 1))) = (♯‘𝑠) ↔ (1...(𝑁 + 1)) ≈ 𝑠))
59	49, 58	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘(1...(𝑁 + 1))) = (♯‘𝑠))
60	12	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
61	59, 60	eqtr3d 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (♯‘𝑠) = (𝑁 + 1))
62	61	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (♯‘𝑠) = (𝑁 + 1))
63	62	oveq2d 7427	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (1...(♯‘𝑠)) = (1...(𝑁 + 1)))
64	63	f1oeq2d 6828	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓:(1...(♯‘𝑠))–1-1-onto→𝑠 ↔ 𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠))
65	56, 64	mpbid 231	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠)
66		f1of1 6831	. . . . . . . 8 ⊢ (𝑓:(1...(𝑁 + 1))–1-1-onto→𝑠 → 𝑓:(1...(𝑁 + 1))–1-1→𝑠)
67	65, 66	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1→𝑠)
68		simplrr 774	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑠 ⊆ 𝐴)
69		f1ss 6792	. . . . . . 7 ⊢ ((𝑓:(1...(𝑁 + 1))–1-1→𝑠 ∧ 𝑠 ⊆ 𝐴) → 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
70	67, 68, 69	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓:(1...(𝑁 + 1))–1-1→𝐴)
71		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠))
72		f1ofo 6839	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝑠))–1-1-onto→𝑠 → 𝑓:(1...(♯‘𝑠))–onto→𝑠)
73		forn 6807	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝑠))–onto→𝑠 → ran 𝑓 = 𝑠)
74		isoeq5 7320	. . . . . . . . 9 ⊢ (ran 𝑓 = 𝑠 → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)))
75	56, 72, 73, 74	4syl 19	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓))
77		isoeq4 7319	. . . . . . . 8 ⊢ ((1...(♯‘𝑠)) = (1...(𝑁 + 1)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), ran 𝑓) ↔ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
78	63, 77	syl 17	. . . . . . 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 231	. . . . . 6 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))
80	70, 79	jca 510	. . . . 5 ⊢ (((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) ∧ 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠)) → (𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
81	80	ex 411	. . . 4 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → (𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
82	81	eximdv 1918	. . 3 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑠)), 𝑠) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))))
83	54, 82	mpd 15	. 2 ⊢ ((𝜑 ∧ ((1...(𝑁 + 1)) ≈ 𝑠 ∧ 𝑠 ⊆ 𝐴)) → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
84	41, 83	exlimddv 1936	1 ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 Vcvv 3472 ⊆ wss 3947 class class class wbr 5147 Or wor 5586 ran crn 5676 –1-1→wf1 6539 –onto→wfo 6540 –1-1-onto→wf1o 6541 ‘cfv 6542 Isom wiso 6543 (class class class)co 7411 ≈ cen 8938 ≼ cdom 8939 Fincfn 8941 ℝcr 11111 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 ℕ₀cn0 12476 ...cfz 13488 ♯chash 14294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-hash 14295

This theorem is referenced by: erdsze2 34494

W3C validator