Theorem peano4nninf 17023

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 17022	. 2 ⊢ 𝑆:ℕ∞⟶ℕ∞
3		fveq1 5692	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4		fveq1 5692	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑥 → (𝑓‘𝑗) = (𝑥‘𝑗))
5	3, 4	sseq12d 3279	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
6	5	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥‘𝑗)))
7		df-nninf 7454	. . . . . . . . 9 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
8	6, 7	elrab2 2985	. . . . . . . 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 6946	. . . . . . 7 ⊢ (𝑥 ∈ (2o ↑𝑚 ω) → 𝑥 Fn ω)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝑥 ∈ ℕ∞ → 𝑥 Fn ω)
12	11	ad2antrr 492	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑥 Fn ω)
13		fveq1 5692	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14		fveq1 5692	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑦 → (𝑓‘𝑗) = (𝑦‘𝑗))
15	13, 14	sseq12d 3279	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
16	15	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦‘𝑗)))
17	16, 7	elrab2 2985	. . . . . . . 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 6946	. . . . . . 7 ⊢ (𝑦 ∈ (2o ↑𝑚 ω) → 𝑦 Fn ω)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝑦 ∈ ℕ∞ → 𝑦 Fn ω)
21	20	ad2antlr 493	. . . . 5 ⊢ (((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) → 𝑦 Fn ω)
22		simplr 533	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑥) = (𝑆‘𝑦))
23	22	fveq1d 5695	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = ((𝑆‘𝑦)‘suc 𝑘))
24		fveq1 5692	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑥 → (𝑝‘∪ 𝑖) = (𝑥‘∪ 𝑖))
25	24	ifeq2d 3659	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑥 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖)))
26	25	mpteq2dv 4220	. . . . . . . . . 10 ⊢ (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
27		omex 4738	. . . . . . . . . . 11 ⊢ ω ∈ V
28	27	mptex 5937	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))) ∈ V
29	26, 1, 28	fvmpt 5779	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ∞ → (𝑆‘𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥‘∪ 𝑖))))
30	29	ad3antrrr 496	. . . . . . . 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 2247	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
33	31	unieqd 3944	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → ∪ 𝑖 = ∪ suc 𝑘)
34	33	fveq2d 5697	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥‘∪ 𝑖) = (𝑥‘∪ suc 𝑘))
35	32, 34	ifbieq2d 3665	. . . . . . . 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 4740	. . . . . . . . 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 6709	. . . . . . . . . 10 ⊢ 1o ∈ 2o
39	38	a1i 9	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40		nninff 7456	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ∞ → 𝑥:ω⟶2o)
41	40	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2o)
42		nnpredcl 4768	. . . . . . . . . . 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 5838	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘∪ suc 𝑘) ∈ 2o)
45		nndceq0 4763	. . . . . . . . . 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 3678	. . . . . . . 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 5783	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)))
49		peano3 4741	. . . . . . . . . 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 2441	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
52	51	iffalsed 3650	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥‘∪ suc 𝑘)) = (𝑥‘∪ suc 𝑘))
53		nnord 4757	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ω → Ord 𝑘)
54		ordtr 4521	. . . . . . . . . . 11 ⊢ (Ord 𝑘 → Tr 𝑘)
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → Tr 𝑘)
56		unisucg 4557	. . . . . . . . . 10 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
57	55, 56	mpbid 147	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
58	57	fveq2d 5697	. . . . . . . 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 2275	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑥)‘suc 𝑘) = (𝑥‘𝑘))
61		fveq1 5692	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑦 → (𝑝‘∪ 𝑖) = (𝑦‘∪ 𝑖))
62	61	ifeq2d 3659	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑦 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖)))
63	62	mpteq2dv 4220	. . . . . . . . . 10 ⊢ (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
64	27	mptex 5937	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))) ∈ V
65	63, 1, 64	fvmpt 5779	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ∞ → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
66	65	ad3antlr 497	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑆‘𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦‘∪ 𝑖))))
67	33	fveq2d 5697	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦‘∪ 𝑖) = (𝑦‘∪ suc 𝑘))
68	32, 67	ifbieq2d 3665	. . . . . . . 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 7456	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ∞ → 𝑦:ω⟶2o)
70	69	ad3antlr 497	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2o)
71	70, 43	ffvelcdmd 5838	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑦‘∪ suc 𝑘) ∈ 2o)
72	39, 71, 46	ifcldcd 3678	. . . . . . . 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 5783	. . . . . . 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 3650	. . . . . . 7 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦‘∪ suc 𝑘)) = (𝑦‘∪ suc 𝑘))
75	57	fveq2d 5697	. . . . . . . 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 2275	. . . . . 6 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆‘𝑦)‘suc 𝑘) = (𝑦‘𝑘))
78	23, 60, 77	3eqtr3d 2279	. . . . 5 ⊢ ((((𝑥 ∈ ℕ∞ ∧ 𝑦 ∈ ℕ∞) ∧ (𝑆‘𝑥) = (𝑆‘𝑦)) ∧ 𝑘 ∈ ω) → (𝑥‘𝑘) = (𝑦‘𝑘))
79	12, 21, 78	eqfnfvd 5803	. . . 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 2604	. 2 ⊢ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)
82		dff13 5968	. 2 ⊢ (𝑆:ℕ∞–1-1→ℕ∞ ↔ (𝑆:ℕ∞⟶ℕ∞ ∧ ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)))
83	2, 81, 82	mpbir2an 955	1 ⊢ 𝑆:ℕ∞–1-1→ℕ∞

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   ⊆ wss 3220  ∅c0 3520  ifcif 3638  ∪ cuni 3933   ↦ cmpt 4190  Tr wtr 4227  Ord word 4505  suc csuc 4508  ωcom 4735   Fn wfn 5370  ⟶wf 5371  –1-1→wf1 5372  ‘cfv 5375  (class class class)co 6079  1_oc1o 6674  2_oc2o 6675   ↑_𝑚 cmap 6916  ℕ_∞xnninf 7453

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1o 6681  df-2o 6682  df-map 6918  df-nninf 7454

This theorem is referenced by:  exmidsbthrlem  17041