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Theorem ssnn0fi 12767
Description: A subset of the nonnegative integers is finite if and only if there is a nonnegative integer so that all integers greater than this integer are not contained in the subset. (Contributed by AV, 3-Oct-2019.)
Assertion
Ref Expression
ssnn0fi (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
Distinct variable group:   𝑆,𝑠,𝑥

Proof of Theorem ssnn0fi
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nn0 11292 . . . . . 6 0 ∈ ℕ0
21a1i 11 . . . . 5 (𝑆 = ∅ → 0 ∈ ℕ0)
3 breq1 4647 . . . . . . . 8 (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
43imbi1d 331 . . . . . . 7 (𝑠 = 0 → ((𝑠 < 𝑥𝑥𝑆) ↔ (0 < 𝑥𝑥𝑆)))
54ralbidv 2983 . . . . . 6 (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆)))
65adantl 482 . . . . 5 ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆)))
7 nnel 2903 . . . . . . . . 9 𝑥𝑆𝑥𝑆)
8 n0i 3912 . . . . . . . . 9 (𝑥𝑆 → ¬ 𝑆 = ∅)
97, 8sylbi 207 . . . . . . . 8 𝑥𝑆 → ¬ 𝑆 = ∅)
109con4i 113 . . . . . . 7 (𝑆 = ∅ → 𝑥𝑆)
1110a1d 25 . . . . . 6 (𝑆 = ∅ → (0 < 𝑥𝑥𝑆))
1211ralrimivw 2964 . . . . 5 (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆))
132, 6, 12rspcedvd 3312 . . . 4 (𝑆 = ∅ → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))
14132a1d 26 . . 3 (𝑆 = ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
15 ltso 10103 . . . . . . 7 < Or ℝ
16 id 22 . . . . . . . . 9 (𝑆 ⊆ ℕ0𝑆 ⊆ ℕ0)
17 nn0ssre 11281 . . . . . . . . 9 0 ⊆ ℝ
1816, 17syl6ss 3607 . . . . . . . 8 (𝑆 ⊆ ℕ0𝑆 ⊆ ℝ)
19183anim3i 1248 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20 fisup2g 8359 . . . . . . 7 (( < Or ℝ ∧ (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ)) → ∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)))
2115, 19, 20sylancr 694 . . . . . 6 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)))
22 simp3 1061 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23 breq2 4648 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (𝑠 < 𝑦𝑠 < 𝑥))
2423notbid 308 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
2524rspcva 3302 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑆 ∧ ∀𝑦𝑆 ¬ 𝑠 < 𝑦) → ¬ 𝑠 < 𝑥)
26252a1d 26 . . . . . . . . . . . . . . . . 17 ((𝑥𝑆 ∧ ∀𝑦𝑆 ¬ 𝑠 < 𝑦) → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ¬ 𝑠 < 𝑥)))
2726expcom 451 . . . . . . . . . . . . . . . 16 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (𝑥𝑆 → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ¬ 𝑠 < 𝑥))))
2827com24 95 . . . . . . . . . . . . . . 15 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → (𝑥 ∈ ℕ0 → (𝑥𝑆 → ¬ 𝑠 < 𝑥))))
2928imp31 448 . . . . . . . . . . . . . 14 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥𝑆 → ¬ 𝑠 < 𝑥))
307, 29syl5bi 232 . . . . . . . . . . . . 13 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (¬ 𝑥𝑆 → ¬ 𝑠 < 𝑥))
3130con4d 114 . . . . . . . . . . . 12 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥𝑥𝑆))
3231ralrimiva 2963 . . . . . . . . . . 11 ((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))
3332ex 450 . . . . . . . . . 10 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3433adantr 481 . . . . . . . . 9 ((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)) → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3534com12 32 . . . . . . . 8 (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3635reximdva 3014 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)) → ∃𝑠𝑆𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
37 ssrexv 3659 . . . . . . 7 (𝑆 ⊆ ℕ0 → (∃𝑠𝑆𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3822, 36, 37sylsyld 61 . . . . . 6 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3921, 38mpd 15 . . . . 5 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))
40393exp 1262 . . . 4 (𝑆 ∈ Fin → (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
4140com3l 89 . . 3 (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
4214, 41pm2.61ine 2874 . 2 (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
43 fzfi 12754 . . . . 5 (0...𝑠) ∈ Fin
44 elfz2nn0 12415 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠))
4544notbii 310 . . . . . . . . . 10 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠))
46 3ianor 1053 . . . . . . . . . 10 (¬ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠))
47 3orass 1039 . . . . . . . . . 10 ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)))
4845, 46, 473bitri 286 . . . . . . . . 9 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)))
49 ssel 3589 . . . . . . . . . . . . 13 (𝑆 ⊆ ℕ0 → (𝑦𝑆𝑦 ∈ ℕ0))
5049adantr 481 . . . . . . . . . . . 12 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (𝑦𝑆𝑦 ∈ ℕ0))
5150adantr 481 . . . . . . . . . . 11 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦𝑆𝑦 ∈ ℕ0))
5251con3rr3 151 . . . . . . . . . 10 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
53 notnotb 304 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54 pm2.24 121 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦𝑆))
5554adantl 482 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦𝑆))
5655adantr 481 . . . . . . . . . . . . . . 15 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦𝑆))
5756com12 32 . . . . . . . . . . . . . 14 𝑠 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
5857a1d 25 . . . . . . . . . . . . 13 𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
59 breq2 4648 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝑠 < 𝑥𝑠 < 𝑦))
60 neleq1 2899 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
6159, 60imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ((𝑠 < 𝑥𝑥𝑆) ↔ (𝑠 < 𝑦𝑦𝑆)))
6261rspcva 3302 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑠 < 𝑦𝑦𝑆))
63 nn0re 11286 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠 ∈ ℕ0𝑠 ∈ ℝ)
64 nn0re 11286 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
65 ltnle 10102 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦𝑠))
6663, 64, 65syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦𝑠))
67 df-nel 2895 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝑆 ↔ ¬ 𝑦𝑆)
6867a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑦𝑆 ↔ ¬ 𝑦𝑆))
6966, 68imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) ↔ (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7069biimpd 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7170ex 450 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7271adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7372com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7473adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7562, 74mpid 44 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7675ex 450 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7776com13 88 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → (𝑦 ∈ ℕ0 → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7877imp 445 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦 ∈ ℕ0 → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7978com13 88 . . . . . . . . . . . . 13 𝑦𝑠 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8058, 79jaoi 394 . . . . . . . . . . . 12 ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8153, 80syl5bir 233 . . . . . . . . . . 11 ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8281impcom 446 . . . . . . . . . 10 ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)) → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
8352, 82jaoi3 1010 . . . . . . . . 9 ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)) → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
8448, 83sylbi 207 . . . . . . . 8 𝑦 ∈ (0...𝑠) → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
8584com12 32 . . . . . . 7 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (¬ 𝑦 ∈ (0...𝑠) → ¬ 𝑦𝑆))
8685con4d 114 . . . . . 6 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦𝑆𝑦 ∈ (0...𝑠)))
8786ssrdv 3601 . . . . 5 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → 𝑆 ⊆ (0...𝑠))
88 ssfi 8165 . . . . 5 (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
8943, 87, 88sylancr 694 . . . 4 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → 𝑆 ∈ Fin)
9089ex 450 . . 3 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → 𝑆 ∈ Fin))
9190rexlimdva 3027 . 2 (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → 𝑆 ∈ Fin))
9242, 91impbid 202 1 (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wnel 2894  wral 2909  wrex 2910  wss 3567  c0 3907   class class class wbr 4644   Or wor 5024  (class class class)co 6635  Fincfn 7940  cr 9920  0cc0 9921   < clt 10059  cle 10060  0cn0 11277  ...cfz 12311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312
This theorem is referenced by:  rabssnn0fi  12768
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