Theorem hashnexinjle 42621

Description: If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. Also we introduce a one sided inequality to simplify a duplicateable proof. (Contributed by metakunt, 2-May-2025.)

Hypotheses

Ref	Expression
hashnexinjle.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashnexinjle.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashnexinjle.3	⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))
hashnexinjle.4	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
hashnexinjle.5	⊢ (𝜑 → 𝐴 ⊆ ℝ)

Assertion

Ref	Expression
hashnexinjle	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem hashnexinjle

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
2		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
3	2	eqeq2d 2751	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
4		breq2 5083	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑧))
5	3, 4	anbi12d 638	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧)))
6		fveqeq2 6843	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑧)))
7		breq1 5082	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑤 < 𝑧))
8	6, 7	anbi12d 638	. . . . . 6 ⊢ (𝑦 = 𝑤 → (((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧)))
9	5, 8	cbvrex2vw 3223	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧))
10	9	bilani 505	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧))
11		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
12	11	eqeq2d 2751	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑧) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
13		breq2 5083	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑤 < 𝑧 ↔ 𝑤 < 𝑦))
14	12, 13	anbi12d 638	. . . . 5 ⊢ (𝑧 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦)))
15		fveqeq2 6843	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
16		breq1 5082	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 < 𝑦 ↔ 𝑥 < 𝑦))
17	15, 16	anbi12d 638	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) ∧ 𝑤 < 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)))
18	14, 17	cbvrex2vw 3223	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤 < 𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
19	10, 18	sylib 219	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
20		rexcom 3269	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
21	19, 20	sylib 219	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
22		hashnexinjle.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
23		hashnexinjle.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
24		hashnexinjle.3	. . . 4 ⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))
25		hashnexinjle.4	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
26	22, 23, 24, 25	hashnexinj 42620	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))
27		simplrl 782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) = (𝐹‘𝑦))
28		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
29	27, 28	jca 516	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
30	29	orcd 879	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
31		simplrl 782	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (𝐹‘𝑥) = (𝐹‘𝑦))
32	31	eqcomd 2746	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
33		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
34	32, 33	jca 516	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))
35	34	olcd 880	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) ∧ 𝑦 < 𝑥) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
36		simprr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
37		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝜑)
38		simprl 776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
39	37, 38	jca 516	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝜑 ∧ 𝑥 ∈ 𝐴))
40		hashnexinjle.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
41	40	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
42	39, 41	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
43	42	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ℝ)
44		simprr 778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
45	37, 44	jca 516	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝜑 ∧ 𝑦 ∈ 𝐴))
46	40	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
47	45, 46	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
48	47	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ℝ)
49	43, 48	lttri2d 11283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑥 ≠ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
50	36, 49	mpbid 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))
51	30, 35, 50	mpjaodan 966	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
52	51	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
53	52	reximdvva 3188	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
54	53	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
55		r19.43 3108	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
56	55	rexbii 3087	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
57	54, 56	sylib 219	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
58		r19.43 3108	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
59	57, 58	sylib 219	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
60	59	ex 413	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥))))
61	26, 60	mpd 15	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦) ∨ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑥) ∧ 𝑦 < 𝑥)))
62	1, 21, 61	mpjaodan 966	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064   ⊆ wss 3890   class class class wbr 5079  ⟶wf 6488  ‘cfv 6492  Fincfn 8890  ℝcr 11035   < clt 11177  ♯chash 14290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-fz 13460  df-hash 14291

This theorem is referenced by:  aks6d1c2  42622