Theorem peano4nninf 16923

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 16922	. 2 ⊢ 𝑆:ℕ∞⟶ℕ∞
3		fveq1 5674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4		fveq1 5674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘𝑗) = (𝑥‘𝑗))
5	3, 4	sseq12d 3273	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
6	5	ralbidv 2544	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
7		df-nninf 7424	. . . . . . . . 9 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
8	6, 7	elrab2 2979	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ∞ ↔ (𝑥 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
9	8	simplbi 274	. . . . . . 7 ⊢ (𝑥 ∈ ℕ∞ → 𝑥 ∈ (2o ↑𝑚 ω))
10		elmapfn 6918	. . . . . . 7 ⊢ (𝑥 ∈ (2o ↑𝑚 ω) → 𝑥 Fn ω)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝑥 ∈ ℕ∞ → 𝑥 Fn ω)
12	11	ad2antrr 488	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑥 Fn ω)
13		fveq1 5674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14		fveq1 5674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘𝑗) = (𝑦‘𝑗))
15	13, 14	sseq12d 3273	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
16	15	ralbidv 2544	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
17	16, 7	elrab2 2979	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ∞ ↔ (𝑦 ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
18	17	simplbi 274	. . . . . . 7 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 ∈ (2o ↑𝑚 ω))
19		elmapfn 6918	. . . . . . 7 ⊢ (𝑦 ∈ (2o ↑𝑚 ω) → 𝑦 Fn ω)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 Fn ω)
21	20	ad2antlr 489	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑦 Fn ω)
22		simplr 529	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑆‘𝑦))
23	22	fveq1d 5677	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = ((𝑆‘𝑦)‘suc 𝑘))
24		fveq1 5674	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑥 → (𝑝‘∪ 𝑖) = (𝑥‘∪ 𝑖))
25	24	ifeq2d 3645	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖)))
26	25	mpteq2dv 4206	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
27		omex 4720	. . . . . . . . . . 11 ⊢ ω ∈ V
28	27	mptex 5917	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))) ∈ V
29	26, 1, 28	fvmpt 5759	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ∞ → (𝑆‘𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
30	29	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
31		simpr 110	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
32	31	eqeq1d 2243	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
33	31	unieqd 3930	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → ∪ 𝑖 = ∪ suc 𝑘)
34	33	fveq2d 5679	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥‘∪ 𝑖) = (𝑥‘∪ suc 𝑘))
35	32, 34	ifbieq2d 3651	. . . . . . . 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 4722	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
37	36	adantl 277	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
38		1lt2o 6688	. . . . . . . . . 10 ⊢ 1o ∈ 2o
39	38	a1i 9	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40		nninff 7426	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
41	40	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2o)
42		nnpredcl 4750	. . . . . . . . . . 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	ffvelcdmd 5818	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) ∈ 2o)
45		nndceq0 4745	. . . . . . . . . 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 3664	. . . . . . . 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 5763	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)))
49		peano3 4723	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → suc 𝑘 ≠ ∅)
50	49	adantl 277	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ≠ ∅)
51	50	neneqd 2435	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
52	51	iffalsed 3636	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)) = (𝑥‘∪ suc 𝑘))
53		nnord 4739	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → Ord 𝑘)
54		ordtr 4504	. . . . . . . . . . 11 ⊢ (Ord 𝑘 → Tr 𝑘)
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → Tr 𝑘)
56		unisucg 4540	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
57	55, 56	mpbid 147	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
58	57	fveq2d 5679	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝑥‘∪ suc 𝑘) = (𝑥‘𝑘))
59	58	adantl 277	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) = (𝑥‘𝑘))
60	48, 52, 59	3eqtrd 2271	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = (𝑥‘𝑘))
61		fveq1 5674	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑦 → (𝑝‘∪ 𝑖) = (𝑦‘∪ 𝑖))
62	61	ifeq2d 3645	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑦 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖)))
63	62	mpteq2dv 4206	. . . . . . . . . 10 ⊢ (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
64	27	mptex 5917	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))) ∈ V
65	63, 1, 64	fvmpt 5759	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ∞ → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
66	65	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
67	33	fveq2d 5679	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦‘∪ 𝑖) = (𝑦‘∪ suc 𝑘))
68	32, 67	ifbieq2d 3651	. . . . . . . 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 7426	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
70	69	ad3antlr 493	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2o)
71	70, 43	ffvelcdmd 5818	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) ∈ 2o)
72	39, 71, 46	ifcldcd 3664	. . . . . . . 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 5763	. . . . . . 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 3636	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)) = (𝑦‘∪ suc 𝑘))
75	57	fveq2d 5679	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝑦‘∪ suc 𝑘) = (𝑦‘𝑘))
76	75	adantl 277	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) = (𝑦‘𝑘))
77	73, 74, 76	3eqtrd 2271	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑦)‘suc 𝑘) = (𝑦‘𝑘))
78	23, 60, 77	3eqtr3d 2275	. . . . 5 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘𝑘) = (𝑦‘𝑘))
79	12, 21, 78	eqfnfvd 5783	. . . 4 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑥 = 𝑦)
80	79	ex 115	. . 3 ⊢ ((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) → ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦))
81	80	rgen2a 2598	. 2 ⊢ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)
82		dff13 5947	. 2 ⊢ (𝑆:ℕ∞–1-1→ℕ∞ ↔ (𝑆:ℕ∞⟶ℕ∞ ∧ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)))
83	2, 81, 82	mpbir2an 951	1 ⊢ 𝑆:ℕ∞–1-1→ℕ∞

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522   ⊆ wss 3214  ∅c0 3512  ifcif 3624  ∪ cuni 3919   ↦ cmpt 4176  Tr wtr 4213  Ord word 4488  suc csuc 4491  ωcom 4717   Fn wfn 5352  ⟶wf 5353  –1-1→wf1 5354  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  2_oc2o 6654   ↑_𝑚 cmap 6895  ℕ_∞xnninf 7423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1o 6660  df-2o 6661  df-map 6897  df-nninf 7424

This theorem is referenced by:  exmidsbthrlem  16941