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Theorem fimaxre2 11249
Description: A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
Assertion
Ref Expression
fimaxre2 ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem fimaxre2
Dummy variables 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3190 . . . 4 (𝑤 = ∅ → (𝑤 ⊆ ℝ ↔ ∅ ⊆ ℝ))
2 raleq 2683 . . . . 5 (𝑤 = ∅ → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦 ∈ ∅ 𝑦𝑥))
32rexbidv 2488 . . . 4 (𝑤 = ∅ → (∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ∅ 𝑦𝑥))
41, 3imbi12d 234 . . 3 (𝑤 = ∅ → ((𝑤 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥) ↔ (∅ ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ∅ 𝑦𝑥)))
5 sseq1 3190 . . . 4 (𝑤 = 𝑢 → (𝑤 ⊆ ℝ ↔ 𝑢 ⊆ ℝ))
6 raleq 2683 . . . . 5 (𝑤 = 𝑢 → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦𝑢 𝑦𝑥))
76rexbidv 2488 . . . 4 (𝑤 = 𝑢 → (∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥))
85, 7imbi12d 234 . . 3 (𝑤 = 𝑢 → ((𝑤 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥) ↔ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)))
9 sseq1 3190 . . . 4 (𝑤 = (𝑢 ∪ {𝑣}) → (𝑤 ⊆ ℝ ↔ (𝑢 ∪ {𝑣}) ⊆ ℝ))
10 raleq 2683 . . . . 5 (𝑤 = (𝑢 ∪ {𝑣}) → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥))
1110rexbidv 2488 . . . 4 (𝑤 = (𝑢 ∪ {𝑣}) → (∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥))
129, 11imbi12d 234 . . 3 (𝑤 = (𝑢 ∪ {𝑣}) → ((𝑤 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥) ↔ ((𝑢 ∪ {𝑣}) ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥)))
13 sseq1 3190 . . . 4 (𝑤 = 𝐴 → (𝑤 ⊆ ℝ ↔ 𝐴 ⊆ ℝ))
14 raleq 2683 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
1514rexbidv 2488 . . . 4 (𝑤 = 𝐴 → (∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥))
1613, 15imbi12d 234 . . 3 (𝑤 = 𝐴 → ((𝑤 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑤 𝑦𝑥) ↔ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)))
17 0re 7971 . . . . 5 0 ∈ ℝ
18 ral0 3536 . . . . 5 𝑦 ∈ ∅ 𝑦 ≤ 0
19 breq2 4019 . . . . . . 7 (𝑥 = 0 → (𝑦𝑥𝑦 ≤ 0))
2019ralbidv 2487 . . . . . 6 (𝑥 = 0 → (∀𝑦 ∈ ∅ 𝑦𝑥 ↔ ∀𝑦 ∈ ∅ 𝑦 ≤ 0))
2120rspcev 2853 . . . . 5 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ∅ 𝑦 ≤ 0) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ∅ 𝑦𝑥)
2217, 18, 21mp2an 426 . . . 4 𝑥 ∈ ℝ ∀𝑦 ∈ ∅ 𝑦𝑥
2322a1i 9 . . 3 (∅ ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ∅ 𝑦𝑥)
24 unss 3321 . . . . . . . . . 10 ((𝑢 ⊆ ℝ ∧ {𝑣} ⊆ ℝ) ↔ (𝑢 ∪ {𝑣}) ⊆ ℝ)
2524biimpri 133 . . . . . . . . 9 ((𝑢 ∪ {𝑣}) ⊆ ℝ → (𝑢 ⊆ ℝ ∧ {𝑣} ⊆ ℝ))
2625simpld 112 . . . . . . . 8 ((𝑢 ∪ {𝑣}) ⊆ ℝ → 𝑢 ⊆ ℝ)
2726adantl 277 . . . . . . 7 (((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) → 𝑢 ⊆ ℝ)
28 simplr 528 . . . . . . 7 (((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) → (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥))
2927, 28mpd 13 . . . . . 6 (((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)
30 breq2 4019 . . . . . . . 8 (𝑥 = 𝑠 → (𝑦𝑥𝑦𝑠))
3130ralbidv 2487 . . . . . . 7 (𝑥 = 𝑠 → (∀𝑦𝑢 𝑦𝑥 ↔ ∀𝑦𝑢 𝑦𝑠))
3231cbvrexv 2716 . . . . . 6 (∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥 ↔ ∃𝑠 ∈ ℝ ∀𝑦𝑢 𝑦𝑠)
3329, 32sylib 122 . . . . 5 (((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) → ∃𝑠 ∈ ℝ ∀𝑦𝑢 𝑦𝑠)
34 simprl 529 . . . . . . 7 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → 𝑠 ∈ ℝ)
3525simprd 114 . . . . . . . . 9 ((𝑢 ∪ {𝑣}) ⊆ ℝ → {𝑣} ⊆ ℝ)
36 vex 2752 . . . . . . . . . 10 𝑣 ∈ V
3736snss 3739 . . . . . . . . 9 (𝑣 ∈ ℝ ↔ {𝑣} ⊆ ℝ)
3835, 37sylibr 134 . . . . . . . 8 ((𝑢 ∪ {𝑣}) ⊆ ℝ → 𝑣 ∈ ℝ)
3938ad2antlr 489 . . . . . . 7 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → 𝑣 ∈ ℝ)
40 maxcl 11233 . . . . . . 7 ((𝑠 ∈ ℝ ∧ 𝑣 ∈ ℝ) → sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ)
4134, 39, 40syl2anc 411 . . . . . 6 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ)
42 nfv 1538 . . . . . . . . . . 11 𝑦 𝑢 ∈ Fin
43 nfv 1538 . . . . . . . . . . . 12 𝑦 𝑢 ⊆ ℝ
44 nfcv 2329 . . . . . . . . . . . . 13 𝑦
45 nfra1 2518 . . . . . . . . . . . . 13 𝑦𝑦𝑢 𝑦𝑥
4644, 45nfrexxy 2526 . . . . . . . . . . . 12 𝑦𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥
4743, 46nfim 1582 . . . . . . . . . . 11 𝑦(𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)
4842, 47nfan 1575 . . . . . . . . . 10 𝑦(𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥))
49 nfv 1538 . . . . . . . . . 10 𝑦(𝑢 ∪ {𝑣}) ⊆ ℝ
5048, 49nfan 1575 . . . . . . . . 9 𝑦((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ)
51 nfv 1538 . . . . . . . . . 10 𝑦 𝑠 ∈ ℝ
52 nfra1 2518 . . . . . . . . . 10 𝑦𝑦𝑢 𝑦𝑠
5351, 52nfan 1575 . . . . . . . . 9 𝑦(𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)
5450, 53nfan 1575 . . . . . . . 8 𝑦(((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠))
55 simprr 531 . . . . . . . . . . . 12 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦𝑢 𝑦𝑠)
56 maxle1 11234 . . . . . . . . . . . . 13 ((𝑠 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < ))
5734, 39, 56syl2anc 411 . . . . . . . . . . . 12 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → 𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < ))
58 r19.27av 2622 . . . . . . . . . . . 12 ((∀𝑦𝑢 𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )) → ∀𝑦𝑢 (𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )))
5955, 57, 58syl2anc 411 . . . . . . . . . . 11 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦𝑢 (𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )))
6059r19.21bi 2575 . . . . . . . . . 10 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → (𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )))
6127ad2antrr 488 . . . . . . . . . . . 12 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → 𝑢 ⊆ ℝ)
62 simpr 110 . . . . . . . . . . . 12 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → 𝑦𝑢)
6361, 62sseldd 3168 . . . . . . . . . . 11 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → 𝑦 ∈ ℝ)
6434adantr 276 . . . . . . . . . . 11 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → 𝑠 ∈ ℝ)
6541adantr 276 . . . . . . . . . . 11 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ)
66 letr 8054 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ) → ((𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )) → 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
6763, 64, 65, 66syl3anc 1248 . . . . . . . . . 10 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → ((𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )) → 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
6860, 67mpd 13 . . . . . . . . 9 (((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) ∧ 𝑦𝑢) → 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
6968ex 115 . . . . . . . 8 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → (𝑦𝑢𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
7054, 69ralrimi 2558 . . . . . . 7 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦𝑢 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
71 maxle2 11235 . . . . . . . . 9 ((𝑠 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < ))
7234, 39, 71syl2anc 411 . . . . . . . 8 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < ))
73 breq1 4018 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ) ↔ 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < )))
7473ralsng 3644 . . . . . . . . 9 (𝑣 ∈ ℝ → (∀𝑦 ∈ {𝑣}𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ) ↔ 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < )))
7539, 74syl 14 . . . . . . . 8 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → (∀𝑦 ∈ {𝑣}𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ) ↔ 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < )))
7672, 75mpbird 167 . . . . . . 7 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦 ∈ {𝑣}𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
77 ralun 3329 . . . . . . 7 ((∀𝑦𝑢 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ) ∧ ∀𝑦 ∈ {𝑣}𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )) → ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
7870, 76, 77syl2anc 411 . . . . . 6 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
79 breq2 4019 . . . . . . . 8 (𝑥 = sup({𝑠, 𝑣}, ℝ, < ) → (𝑦𝑥𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
8079ralbidv 2487 . . . . . . 7 (𝑥 = sup({𝑠, 𝑣}, ℝ, < ) → (∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥 ↔ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
8180rspcev 2853 . . . . . 6 ((sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ ∧ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥)
8241, 78, 81syl2anc 411 . . . . 5 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥)
8333, 82rexlimddv 2609 . . . 4 (((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥)
8483exp31 364 . . 3 (𝑢 ∈ Fin → ((𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥) → ((𝑢 ∪ {𝑣}) ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥)))
854, 8, 12, 16, 23, 84findcard2 6903 . 2 (𝐴 ∈ Fin → (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥))
8685impcom 125 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  cun 3139  wss 3141  c0 3434  {csn 3604  {cpr 3605   class class class wbr 4015  Fincfn 6754  supcsup 6995  cr 7824  0cc0 7825   < clt 8006  cle 8007
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-er 6549  df-en 6755  df-fin 6757  df-sup 6997  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-rp 9668  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022
This theorem is referenced by:  fsum3cvg3  11418
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