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Theorem iccsupr 8935
 Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
Assertion
Ref Expression
iccsupr (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem iccsupr
StepHypRef Expression
1 iccssre 8924 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
2 sstr 2980 . . . . 5 ((𝑆 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ) → 𝑆 ⊆ ℝ)
32ancoms 259 . . . 4 (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
41, 3sylan 271 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
543adant3 935 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → 𝑆 ⊆ ℝ)
6 ne0i 3257 . . 3 (𝐶𝑆𝑆 ≠ ∅)
763ad2ant3 938 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → 𝑆 ≠ ∅)
8 simplr 490 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
9 ssel 2966 . . . . . . . 8 (𝑆 ⊆ (𝐴[,]𝐵) → (𝑦𝑆𝑦 ∈ (𝐴[,]𝐵)))
10 elicc2 8907 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
1110biimpd 136 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
129, 11sylan9r 396 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → (𝑦𝑆 → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
1312imp 119 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦𝑆) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
1413simp3d 929 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦𝑆) → 𝑦𝐵)
1514ralrimiva 2409 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∀𝑦𝑆 𝑦𝐵)
16 breq2 3795 . . . . . 6 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
1716ralbidv 2343 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝑆 𝑦𝑥 ↔ ∀𝑦𝑆 𝑦𝐵))
1817rspcev 2673 . . . 4 ((𝐵 ∈ ℝ ∧ ∀𝑦𝑆 𝑦𝐵) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
198, 15, 18syl2anc 397 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
20193adant3 935 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
215, 7, 203jca 1095 1 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896   = wceq 1259   ∈ wcel 1409   ≠ wne 2220  ∀wral 2323  ∃wrex 2324   ⊆ wss 2944  ∅c0 3251   class class class wbr 3791  (class class class)co 5539  ℝcr 6945   ≤ cle 7119  [,]cicc 8860 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-cnex 7032  ax-resscn 7033  ax-pre-ltirr 7053  ax-pre-ltwlin 7054  ax-pre-lttrn 7055 This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-po 4060  df-iso 4061  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-icc 8864 This theorem is referenced by: (None)
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