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Theorem fimaxre2 11219
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 3178 . . . 4 (𝑤 = ∅ → (𝑤 ⊆ ℝ ↔ ∅ ⊆ ℝ))
2 raleq 2672 . . . . 5 (𝑤 = ∅ → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦 ∈ ∅ 𝑦𝑥))
32rexbidv 2478 . . . 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 3178 . . . 4 (𝑤 = 𝑢 → (𝑤 ⊆ ℝ ↔ 𝑢 ⊆ ℝ))
6 raleq 2672 . . . . 5 (𝑤 = 𝑢 → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦𝑢 𝑦𝑥))
76rexbidv 2478 . . . 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 3178 . . . 4 (𝑤 = (𝑢 ∪ {𝑣}) → (𝑤 ⊆ ℝ ↔ (𝑢 ∪ {𝑣}) ⊆ ℝ))
10 raleq 2672 . . . . 5 (𝑤 = (𝑢 ∪ {𝑣}) → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥))
1110rexbidv 2478 . . . 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 3178 . . . 4 (𝑤 = 𝐴 → (𝑤 ⊆ ℝ ↔ 𝐴 ⊆ ℝ))
14 raleq 2672 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝑤 𝑦𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
1514rexbidv 2478 . . . 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 7948 . . . . 5 0 ∈ ℝ
18 ral0 3524 . . . . 5 𝑦 ∈ ∅ 𝑦 ≤ 0
19 breq2 4004 . . . . . . 7 (𝑥 = 0 → (𝑦𝑥𝑦 ≤ 0))
2019ralbidv 2477 . . . . . 6 (𝑥 = 0 → (∀𝑦 ∈ ∅ 𝑦𝑥 ↔ ∀𝑦 ∈ ∅ 𝑦 ≤ 0))
2120rspcev 2841 . . . . 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 3309 . . . . . . . . . 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 4004 . . . . . . . 8 (𝑥 = 𝑠 → (𝑦𝑥𝑦𝑠))
3130ralbidv 2477 . . . . . . 7 (𝑥 = 𝑠 → (∀𝑦𝑢 𝑦𝑥 ↔ ∀𝑦𝑢 𝑦𝑠))
3231cbvrexv 2704 . . . . . 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 2740 . . . . . . . . . 10 𝑣 ∈ V
3736snss 3726 . . . . . . . . 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 11203 . . . . . . 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 1528 . . . . . . . . . . 11 𝑦 𝑢 ∈ Fin
43 nfv 1528 . . . . . . . . . . . 12 𝑦 𝑢 ⊆ ℝ
44 nfcv 2319 . . . . . . . . . . . . 13 𝑦
45 nfra1 2508 . . . . . . . . . . . . 13 𝑦𝑦𝑢 𝑦𝑥
4644, 45nfrexxy 2516 . . . . . . . . . . . 12 𝑦𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥
4743, 46nfim 1572 . . . . . . . . . . 11 𝑦(𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)
4842, 47nfan 1565 . . . . . . . . . 10 𝑦(𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥))
49 nfv 1528 . . . . . . . . . 10 𝑦(𝑢 ∪ {𝑣}) ⊆ ℝ
5048, 49nfan 1565 . . . . . . . . 9 𝑦((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ)
51 nfv 1528 . . . . . . . . . 10 𝑦 𝑠 ∈ ℝ
52 nfra1 2508 . . . . . . . . . 10 𝑦𝑦𝑢 𝑦𝑠
5351, 52nfan 1565 . . . . . . . . 9 𝑦(𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)
5450, 53nfan 1565 . . . . . . . 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 11204 . . . . . . . . . . . . 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 2612 . . . . . . . . . . . 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 2565 . . . . . . . . . 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 3156 . . . . . . . . . . 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 8030 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ sup({𝑠, 𝑣}, ℝ, < ) ∈ ℝ) → ((𝑦𝑠𝑠 ≤ sup({𝑠, 𝑣}, ℝ, < )) → 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
6763, 64, 65, 66syl3anc 1238 . . . . . . . . . 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 2548 . . . . . . 7 ((((𝑢 ∈ Fin ∧ (𝑢 ⊆ ℝ → ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥)) ∧ (𝑢 ∪ {𝑣}) ⊆ ℝ) ∧ (𝑠 ∈ ℝ ∧ ∀𝑦𝑢 𝑦𝑠)) → ∀𝑦𝑢 𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ))
71 maxle2 11205 . . . . . . . . 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 4003 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < ) ↔ 𝑣 ≤ sup({𝑠, 𝑣}, ℝ, < )))
7473ralsng 3631 . . . . . . . . 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 3317 . . . . . . 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 4004 . . . . . . . 8 (𝑥 = sup({𝑠, 𝑣}, ℝ, < ) → (𝑦𝑥𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
8079ralbidv 2477 . . . . . . 7 (𝑥 = sup({𝑠, 𝑣}, ℝ, < ) → (∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦𝑥 ↔ ∀𝑦 ∈ (𝑢 ∪ {𝑣})𝑦 ≤ sup({𝑠, 𝑣}, ℝ, < )))
8180rspcev 2841 . . . . . 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 2599 . . . 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 6883 . 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 1353  wcel 2148  wral 2455  wrex 2456  cun 3127  wss 3129  c0 3422  {csn 3591  {cpr 3592   class class class wbr 4000  Fincfn 6734  supcsup 6975  cr 7801  0cc0 7802   < clt 7982  cle 7983
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-er 6529  df-en 6735  df-fin 6737  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992
This theorem is referenced by:  fsum3cvg3  11388
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