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Theorem iccsupr 9445
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 9434 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
2 sstr 3034 . . . . 5 ((𝑆 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ) → 𝑆 ⊆ ℝ)
32ancoms 265 . . . 4 (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
41, 3sylan 278 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
543adant3 964 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → 𝑆 ⊆ ℝ)
6 ne0i 3293 . . 3 (𝐶𝑆𝑆 ≠ ∅)
763ad2ant3 967 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → 𝑆 ≠ ∅)
8 simplr 498 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
9 ssel 3020 . . . . . . . 8 (𝑆 ⊆ (𝐴[,]𝐵) → (𝑦𝑆𝑦 ∈ (𝐴[,]𝐵)))
10 elicc2 9417 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
1110biimpd 143 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
129, 11sylan9r 403 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → (𝑦𝑆 → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
1312imp 123 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦𝑆) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
1413simp3d 958 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦𝑆) → 𝑦𝐵)
1514ralrimiva 2447 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∀𝑦𝑆 𝑦𝐵)
16 breq2 3855 . . . . . 6 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
1716ralbidv 2381 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝑆 𝑦𝑥 ↔ ∀𝑦𝑆 𝑦𝐵))
1817rspcev 2723 . . . 4 ((𝐵 ∈ ℝ ∧ ∀𝑦𝑆 𝑦𝐵) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
198, 15, 18syl2anc 404 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
20193adant3 964 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥)
215, 7, 203jca 1124 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 103  w3a 925   = wceq 1290  wcel 1439  wne 2256  wral 2360  wrex 2361  wss 3000  c0 3287   class class class wbr 3851  (class class class)co 5666  cr 7410  cle 7584  [,]cicc 9370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-icc 9374
This theorem is referenced by: (None)
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