Theorem peano4nninf 13189

Description: The successor function on ℕ_∞ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)

Hypothesis

Ref	Expression
peano4nninf.s	⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))

Assertion

Ref	Expression
peano4nninf	⊢ 𝑆:ℕ∞–1-1→ℕ∞

Distinct variable groups:   𝑆,𝑖   𝑖,𝑝

Allowed substitution hint:   𝑆(𝑝)

Proof of Theorem peano4nninf

Dummy variables 𝑘 𝑥 𝑦 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano4nninf.s	. . 3 ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))
2	1	nnsf 13188	. 2 ⊢ 𝑆:ℕ∞⟶ℕ∞
3		fveq1 5413	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4		fveq1 5413	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘𝑗) = (𝑥‘𝑗))
5	3, 4	sseq12d 3123	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
6	5	ralbidv 2435	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
7		df-nninf 7000	. . . . . . . . 9 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
8	6, 7	elrab2 2838	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ∞ ↔ (𝑥 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
9	8	simplbi 272	. . . . . . 7 ⊢ (𝑥 ∈ ℕ∞ → 𝑥 ∈ (2o ↑𝑚 ω))
10		elmapfn 6558	. . . . . . 7 ⊢ (𝑥 ∈ (2o ↑𝑚 ω) → 𝑥 Fn ω)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝑥 ∈ ℕ∞ → 𝑥 Fn ω)
12	11	ad2antrr 479	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑥 Fn ω)
13		fveq1 5413	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14		fveq1 5413	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘𝑗) = (𝑦‘𝑗))
15	13, 14	sseq12d 3123	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
16	15	ralbidv 2435	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
17	16, 7	elrab2 2838	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ∞ ↔ (𝑦 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
18	17	simplbi 272	. . . . . . 7 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 ∈ (2o ↑𝑚 ω))
19		elmapfn 6558	. . . . . . 7 ⊢ (𝑦 ∈ (2o ↑𝑚 ω) → 𝑦 Fn ω)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 Fn ω)
21	20	ad2antlr 480	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑦 Fn ω)
22		simplr 519	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑆‘𝑦))
23	22	fveq1d 5416	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = ((𝑆‘𝑦)‘suc 𝑘))
24		fveq1 5413	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑥 → (𝑝‘∪ 𝑖) = (𝑥‘∪ 𝑖))
25	24	ifeq2d 3485	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖)))
26	25	mpteq2dv 4014	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
27		omex 4502	. . . . . . . . . . 11 ⊢ ω ∈ V
28	27	mptex 5639	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))) ∈ V
29	26, 1, 28	fvmpt 5491	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ∞ → (𝑆‘𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
30	29	ad3antrrr 483	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
31		simpr 109	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
32	31	eqeq1d 2146	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
33	31	unieqd 3742	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → ∪ 𝑖 = ∪ suc 𝑘)
34	33	fveq2d 5418	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥‘∪ 𝑖) = (𝑥‘∪ suc 𝑘))
35	32, 34	ifbieq2d 3491	. . . . . . . 8 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖)) = if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)))
36		peano2 4504	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
37	36	adantl 275	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
38		1lt2o 6332	. . . . . . . . . 10 ⊢ 1o ∈ 2o
39	38	a1i 9	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40		nninff 13187	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
41	40	ad3antrrr 483	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2o)
42		nnpredcl 4531	. . . . . . . . . . 11 ⊢ (suc 𝑘 ∈ ω → ∪ suc 𝑘 ∈ ω)
43	37, 42	syl 14	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ∪ suc 𝑘 ∈ ω)
44	41, 43	ffvelrnd 5549	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) ∈ 2o)
45		nndceq0 4526	. . . . . . . . . 10 ⊢ (suc 𝑘 ∈ ω → DECID suc 𝑘 = ∅)
46	37, 45	syl 14	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → DECID suc 𝑘 = ∅)
47	39, 44, 46	ifcldcd 3502	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)) ∈ 2o)
48	30, 35, 37, 47	fvmptd 5495	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)))
49		peano3 4505	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → suc 𝑘 ≠ ∅)
50	49	adantl 275	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ≠ ∅)
51	50	neneqd 2327	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
52	51	iffalsed 3479	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)) = (𝑥‘∪ suc 𝑘))
53		nnord 4520	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → Ord 𝑘)
54		ordtr 4295	. . . . . . . . . . 11 ⊢ (Ord 𝑘 → Tr 𝑘)
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → Tr 𝑘)
56		unisucg 4331	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
57	55, 56	mpbid 146	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
58	57	fveq2d 5418	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝑥‘∪ suc 𝑘) = (𝑥‘𝑘))
59	58	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) = (𝑥‘𝑘))
60	48, 52, 59	3eqtrd 2174	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = (𝑥‘𝑘))
61		fveq1 5413	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑦 → (𝑝‘∪ 𝑖) = (𝑦‘∪ 𝑖))
62	61	ifeq2d 3485	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑦 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖)))
63	62	mpteq2dv 4014	. . . . . . . . . 10 ⊢ (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
64	27	mptex 5639	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))) ∈ V
65	63, 1, 64	fvmpt 5491	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ∞ → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
66	65	ad3antlr 484	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
67	33	fveq2d 5418	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦‘∪ 𝑖) = (𝑦‘∪ suc 𝑘))
68	32, 67	ifbieq2d 3491	. . . . . . . 8 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖)) = if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)))
69		nninff 13187	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
70	69	ad3antlr 484	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2o)
71	70, 43	ffvelrnd 5549	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) ∈ 2o)
72	39, 71, 46	ifcldcd 3502	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)) ∈ 2o)
73	66, 68, 37, 72	fvmptd 5495	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑦)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)))
74	51	iffalsed 3479	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)) = (𝑦‘∪ suc 𝑘))
75	57	fveq2d 5418	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝑦‘∪ suc 𝑘) = (𝑦‘𝑘))
76	75	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) = (𝑦‘𝑘))
77	73, 74, 76	3eqtrd 2174	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑦)‘suc 𝑘) = (𝑦‘𝑘))
78	23, 60, 77	3eqtr3d 2178	. . . . 5 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘𝑘) = (𝑦‘𝑘))
79	12, 21, 78	eqfnfvd 5514	. . . 4 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑥 = 𝑦)
80	79	ex 114	. . 3 ⊢ ((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) → ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦))
81	80	rgen2a 2484	. 2 ⊢ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)
82		dff13 5662	. 2 ⊢ (𝑆:ℕ∞–1-1→ℕ∞ ↔ (𝑆:ℕ∞⟶ℕ∞ ∧ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)))
83	2, 81, 82	mpbir2an 926	1 ⊢ 𝑆:ℕ∞–1-1→ℕ∞

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 819   = wceq 1331   ∈ wcel 1480   ≠ wne 2306  ∀wral 2414   ⊆ wss 3066  ∅c0 3358  ifcif 3469  ∪ cuni 3731   ↦ cmpt 3984  Tr wtr 4021  Ord word 4279  suc csuc 4282  ωcom 4499   Fn wfn 5113  ⟶wf 5114  –1-1→wf1 5115  ‘cfv 5118  (class class class)co 5767  1_oc1o 6299  2_oc2o 6300   ↑_𝑚 cmap 6535  ℕ_∞xnninf 6998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1o 6306  df-2o 6307  df-map 6537  df-nninf 7000

This theorem is referenced by:  exmidsbthrlem  13206