Theorem peano4nninf 15566

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 15565	. 2 ⊢ 𝑆:ℕ∞⟶ℕ∞
3		fveq1 5554	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4		fveq1 5554	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘𝑗) = (𝑥‘𝑗))
5	3, 4	sseq12d 3211	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
6	5	ralbidv 2494	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
7		df-nninf 7181	. . . . . . . . 9 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
8	6, 7	elrab2 2920	. . . . . . . 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 6727	. . . . . . 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 5554	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14		fveq1 5554	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘𝑗) = (𝑦‘𝑗))
15	13, 14	sseq12d 3211	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
16	15	ralbidv 2494	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
17	16, 7	elrab2 2920	. . . . . . . 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 6727	. . . . . . 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 528	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑆‘𝑦))
23	22	fveq1d 5557	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = ((𝑆‘𝑦)‘suc 𝑘))
24		fveq1 5554	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑥 → (𝑝‘∪ 𝑖) = (𝑥‘∪ 𝑖))
25	24	ifeq2d 3576	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖)))
26	25	mpteq2dv 4121	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
27		omex 4626	. . . . . . . . . . 11 ⊢ ω ∈ V
28	27	mptex 5785	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))) ∈ V
29	26, 1, 28	fvmpt 5635	. . . . . . . . 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 2202	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
33	31	unieqd 3847	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → ∪ 𝑖 = ∪ suc 𝑘)
34	33	fveq2d 5559	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥‘∪ 𝑖) = (𝑥‘∪ suc 𝑘))
35	32, 34	ifbieq2d 3582	. . . . . . . 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 4628	. . . . . . . . 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 6497	. . . . . . . . . 10 ⊢ 1o ∈ 2o
39	38	a1i 9	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40		nninff 7183	. . . . . . . . . . 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 4656	. . . . . . . . . . 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 5695	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) ∈ 2o)
45		nndceq0 4651	. . . . . . . . . 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 3594	. . . . . . . 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 5639	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)))
49		peano3 4629	. . . . . . . . . 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 2385	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
52	51	iffalsed 3568	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)) = (𝑥‘∪ suc 𝑘))
53		nnord 4645	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → Ord 𝑘)
54		ordtr 4410	. . . . . . . . . . 11 ⊢ (Ord 𝑘 → Tr 𝑘)
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → Tr 𝑘)
56		unisucg 4446	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
57	55, 56	mpbid 147	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
58	57	fveq2d 5559	. . . . . . . 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 2230	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = (𝑥‘𝑘))
61		fveq1 5554	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑦 → (𝑝‘∪ 𝑖) = (𝑦‘∪ 𝑖))
62	61	ifeq2d 3576	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑦 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖)))
63	62	mpteq2dv 4121	. . . . . . . . . 10 ⊢ (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
64	27	mptex 5785	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))) ∈ V
65	63, 1, 64	fvmpt 5635	. . . . . . . . 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 5559	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦‘∪ 𝑖) = (𝑦‘∪ suc 𝑘))
68	32, 67	ifbieq2d 3582	. . . . . . . 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 7183	. . . . . . . . . . 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 5695	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) ∈ 2o)
72	39, 71, 46	ifcldcd 3594	. . . . . . . 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 5639	. . . . . . 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 3568	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)) = (𝑦‘∪ suc 𝑘))
75	57	fveq2d 5559	. . . . . . . 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 2230	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑦)‘suc 𝑘) = (𝑦‘𝑘))
78	23, 60, 77	3eqtr3d 2234	. . . . 5 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘𝑘) = (𝑦‘𝑘))
79	12, 21, 78	eqfnfvd 5659	. . . 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 2548	. 2 ⊢ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)
82		dff13 5812	. 2 ⊢ (𝑆:ℕ∞–1-1→ℕ∞ ↔ (𝑆:ℕ∞⟶ℕ∞ ∧ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)))
83	2, 81, 82	mpbir2an 944	1 ⊢ 𝑆:ℕ∞–1-1→ℕ∞

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472   ⊆ wss 3154  ∅c0 3447  ifcif 3558  ∪ cuni 3836   ↦ cmpt 4091  Tr wtr 4128  Ord word 4394  suc csuc 4397  ωcom 4623   Fn wfn 5250  ⟶wf 5251  –1-1→wf1 5252  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  2_oc2o 6465   ↑_𝑚 cmap 6704  ℕ_∞xnninf 7180

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1o 6471  df-2o 6472  df-map 6706  df-nninf 7181

This theorem is referenced by:  exmidsbthrlem  15582