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Theorem ssnn0fi 12597
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 11150 . . . . . 6 0 ∈ ℕ0
21a1i 11 . . . . 5 (𝑆 = ∅ → 0 ∈ ℕ0)
3 breq1 4576 . . . . . . . 8 (𝑠 = 0 → (𝑠 < 𝑥 ↔ 0 < 𝑥))
43imbi1d 329 . . . . . . 7 (𝑠 = 0 → ((𝑠 < 𝑥𝑥𝑆) ↔ (0 < 𝑥𝑥𝑆)))
54ralbidv 2964 . . . . . 6 (𝑠 = 0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆)))
65adantl 480 . . . . 5 ((𝑆 = ∅ ∧ 𝑠 = 0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) ↔ ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆)))
7 nnel 2887 . . . . . . . . 9 𝑥𝑆𝑥𝑆)
8 n0i 3874 . . . . . . . . 9 (𝑥𝑆 → ¬ 𝑆 = ∅)
97, 8sylbi 205 . . . . . . . 8 𝑥𝑆 → ¬ 𝑆 = ∅)
109con4i 111 . . . . . . 7 (𝑆 = ∅ → 𝑥𝑆)
1110a1d 25 . . . . . 6 (𝑆 = ∅ → (0 < 𝑥𝑥𝑆))
1211ralrimivw 2945 . . . . 5 (𝑆 = ∅ → ∀𝑥 ∈ ℕ0 (0 < 𝑥𝑥𝑆))
132, 6, 12rspcedvd 3284 . . . 4 (𝑆 = ∅ → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))
14132a1d 26 . . 3 (𝑆 = ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
15 ltso 9965 . . . . . . 7 < Or ℝ
16 id 22 . . . . . . . . 9 (𝑆 ⊆ ℕ0𝑆 ⊆ ℕ0)
17 nn0ssre 11139 . . . . . . . . 9 0 ⊆ ℝ
1816, 17syl6ss 3575 . . . . . . . 8 (𝑆 ⊆ ℕ0𝑆 ⊆ ℝ)
19183anim3i 1242 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ))
20 fisup2g 8230 . . . . . . 7 (( < Or ℝ ∧ (𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ)) → ∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)))
2115, 19, 20sylancr 693 . . . . . 6 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → ∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)))
22 simp3 1055 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → 𝑆 ⊆ ℕ0)
23 breq2 4577 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (𝑠 < 𝑦𝑠 < 𝑥))
2423notbid 306 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥))
2524rspcva 3275 . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . 16 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (𝑥𝑆 → (𝑥 ∈ ℕ0 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ¬ 𝑠 < 𝑥))))
2827com24 92 . . . . . . . . . . . . . . 15 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → (𝑥 ∈ ℕ0 → (𝑥𝑆 → ¬ 𝑠 < 𝑥))))
2928imp31 446 . . . . . . . . . . . . . 14 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑥𝑆 → ¬ 𝑠 < 𝑥))
307, 29syl5bi 230 . . . . . . . . . . . . 13 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (¬ 𝑥𝑆 → ¬ 𝑠 < 𝑥))
3130con4d 112 . . . . . . . . . . . 12 (((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) ∧ 𝑥 ∈ ℕ0) → (𝑠 < 𝑥𝑥𝑆))
3231ralrimiva 2944 . . . . . . . . . . 11 ((∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆)) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))
3332ex 448 . . . . . . . . . 10 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 → (((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) ∧ 𝑠𝑆) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3433adantr 479 . . . . . . . . 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 2995 . . . . . . 7 ((𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0) → (∃𝑠𝑆 (∀𝑦𝑆 ¬ 𝑠 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑠 → ∃𝑧𝑆 𝑦 < 𝑧)) → ∃𝑠𝑆𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
37 ssrexv 3625 . . . . . . 7 (𝑆 ⊆ ℕ0 → (∃𝑠𝑆𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
3822, 36, 37sylsyld 58 . . . . . 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 1255 . . . 4 (𝑆 ∈ Fin → (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
4140com3l 86 . . 3 (𝑆 ≠ ∅ → (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆))))
4214, 41pm2.61ine 2860 . 2 (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
43 fzfi 12584 . . . . 5 (0...𝑠) ∈ Fin
44 elfz2nn0 12251 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑠) ↔ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠))
4544notbii 308 . . . . . . . . . 10 𝑦 ∈ (0...𝑠) ↔ ¬ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠))
46 3ianor 1047 . . . . . . . . . 10 (¬ (𝑦 ∈ ℕ0𝑠 ∈ ℕ0𝑦𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠))
47 3orass 1033 . . . . . . . . . 10 ((¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)))
4845, 46, 473bitri 284 . . . . . . . . 9 𝑦 ∈ (0...𝑠) ↔ (¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)))
49 ssel 3557 . . . . . . . . . . . . 13 (𝑆 ⊆ ℕ0 → (𝑦𝑆𝑦 ∈ ℕ0))
5049adantr 479 . . . . . . . . . . . 12 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (𝑦𝑆𝑦 ∈ ℕ0))
5150adantr 479 . . . . . . . . . . 11 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦𝑆𝑦 ∈ ℕ0))
5251con3rr3 149 . . . . . . . . . 10 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
53 notnotb 302 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0)
54 pm2.24 119 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ0 → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦𝑆))
5554adantl 480 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑠 ∈ ℕ0 → ¬ 𝑦𝑆))
5655adantr 479 . . . . . . . . . . . . . . 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 4577 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝑠 < 𝑥𝑠 < 𝑦))
60 neleq1 2883 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝑥𝑆𝑦𝑆))
6159, 60imbi12d 332 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → ((𝑠 < 𝑥𝑥𝑆) ↔ (𝑠 < 𝑦𝑦𝑆)))
6261rspcva 3275 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑠 < 𝑦𝑦𝑆))
63 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠 ∈ ℕ0𝑠 ∈ ℝ)
64 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
65 ltnle 9964 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑠 < 𝑦 ↔ ¬ 𝑦𝑠))
6663, 64, 65syl2an 492 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑠 < 𝑦 ↔ ¬ 𝑦𝑠))
67 df-nel 2778 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝑆 ↔ ¬ 𝑦𝑆)
6867a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑦𝑆 ↔ ¬ 𝑦𝑆))
6966, 68imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) ↔ (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7069biimpd 217 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ ℕ0𝑦 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7170ex 448 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℕ0 → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7271adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (𝑦 ∈ ℕ0 → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7372com12 32 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7473adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → ((𝑠 < 𝑦𝑦𝑆) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7562, 74mpid 42 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7675ex 448 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7776com13 85 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → (𝑦 ∈ ℕ0 → (¬ 𝑦𝑠 → ¬ 𝑦𝑆))))
7877imp 443 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦 ∈ ℕ0 → (¬ 𝑦𝑠 → ¬ 𝑦𝑆)))
7978com13 85 . . . . . . . . . . . . 13 𝑦𝑠 → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8058, 79jaoi 392 . . . . . . . . . . . 12 ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) → (𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8153, 80syl5bir 231 . . . . . . . . . . 11 ((¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠) → (¬ ¬ 𝑦 ∈ ℕ0 → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆)))
8281impcom 444 . . . . . . . . . 10 ((¬ ¬ 𝑦 ∈ ℕ0 ∧ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)) → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
8352, 82jaoi3 1002 . . . . . . . . 9 ((¬ 𝑦 ∈ ℕ0 ∨ (¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦𝑠)) → (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → ¬ 𝑦𝑆))
8448, 83sylbi 205 . . . . . . . 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 112 . . . . . 6 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → (𝑦𝑆𝑦 ∈ (0...𝑠)))
8786ssrdv 3569 . . . . 5 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → 𝑆 ⊆ (0...𝑠))
88 ssfi 8038 . . . . 5 (((0...𝑠) ∈ Fin ∧ 𝑆 ⊆ (0...𝑠)) → 𝑆 ∈ Fin)
8943, 87, 88sylancr 693 . . . 4 (((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)) → 𝑆 ∈ Fin)
9089ex 448 . . 3 ((𝑆 ⊆ ℕ0𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → 𝑆 ∈ Fin))
9190rexlimdva 3008 . 2 (𝑆 ⊆ ℕ0 → (∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆) → 𝑆 ∈ Fin))
9242, 91impbid 200 1 (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3o 1029  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wnel 2776  wral 2891  wrex 2892  wss 3535  c0 3869   class class class wbr 4573   Or wor 4944  (class class class)co 6523  Fincfn 7814  cr 9787  0cc0 9788   < clt 9926  cle 9927  0cn0 11135  ...cfz 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149
This theorem is referenced by:  rabssnn0fi  12598
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