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Mirrors > Home > MPE Home > Th. List > suprzcl2 | Structured version Visualization version GIF version |
Description: The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 12400 avoids ax-pre-sup 10949.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
suprzcl2 | ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsupss 12677 | . 2 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
2 | ssel2 3916 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ) | |
3 | 2 | zred 12426 | . . . . . 6 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
4 | ltso 11055 | . . . . . . . . . 10 ⊢ < Or ℝ | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → < Or ℝ) |
6 | 5 | eqsup 9215 | . . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑥)) |
7 | 6 | mptru 1546 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑥) |
8 | 7 | 3expib 1121 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑥)) |
9 | 3, 8 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑥)) |
10 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
11 | eleq1 2826 | . . . . . 6 ⊢ (sup(𝐴, ℝ, < ) = 𝑥 → (sup(𝐴, ℝ, < ) ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
12 | 10, 11 | syl5ibrcom 246 | . . . . 5 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → (sup(𝐴, ℝ, < ) = 𝑥 → sup(𝐴, ℝ, < ) ∈ 𝐴)) |
13 | 9, 12 | syld 47 | . . . 4 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) ∈ 𝐴)) |
14 | 13 | rexlimdva 3213 | . . 3 ⊢ (𝐴 ⊆ ℤ → (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) ∈ 𝐴)) |
15 | 14 | 3ad2ant1 1132 | . 2 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → sup(𝐴, ℝ, < ) ∈ 𝐴)) |
16 | 1, 15 | mpd 15 | 1 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 Or wor 5502 supcsup 9199 ℝcr 10870 < clt 11009 ≤ cle 11010 ℤcz 12319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 |
This theorem is referenced by: suprzub 12679 gcdcllem3 16208 maxprmfct 16414 pcprecl 16540 prmreclem1 16617 0ram 16721 0ramcl 16724 gexex 19454 |
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