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Mirrors > Home > MPE Home > Th. List > supexpr | Structured version Visualization version GIF version |
Description: The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
supexpr | ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suplem1pr 10476 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P) | |
2 | ltrelpr 10422 | . . . . . . . . 9 ⊢ <P ⊆ (P × P) | |
3 | 2 | brel 5619 | . . . . . . . 8 ⊢ (𝑦<P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P)) |
4 | 3 | simpld 497 | . . . . . . 7 ⊢ (𝑦<P 𝑥 → 𝑦 ∈ P) |
5 | 4 | ralimi 3162 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ P) |
6 | dfss3 3958 | . . . . . 6 ⊢ (𝐴 ⊆ P ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ P) | |
7 | 5, 6 | sylibr 236 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P) |
8 | 7 | rexlimivw 3284 | . . . 4 ⊢ (∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P) |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → 𝐴 ⊆ P) |
10 | suplem2pr 10477 | . . . . . 6 ⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) | |
11 | 10 | simpld 497 | . . . . 5 ⊢ (𝐴 ⊆ P → (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦)) |
12 | 11 | ralrimiv 3183 | . . . 4 ⊢ (𝐴 ⊆ P → ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦) |
13 | 10 | simprd 498 | . . . . 5 ⊢ (𝐴 ⊆ P → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)) |
14 | 13 | ralrimivw 3185 | . . . 4 ⊢ (𝐴 ⊆ P → ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)) |
15 | 12, 14 | jca 514 | . . 3 ⊢ (𝐴 ⊆ P → (∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
16 | 9, 15 | syl 17 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → (∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
17 | breq1 5071 | . . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑥<P 𝑦 ↔ ∪ 𝐴<P 𝑦)) | |
18 | 17 | notbid 320 | . . . . 5 ⊢ (𝑥 = ∪ 𝐴 → (¬ 𝑥<P 𝑦 ↔ ¬ ∪ 𝐴<P 𝑦)) |
19 | 18 | ralbidv 3199 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦)) |
20 | breq2 5072 | . . . . . 6 ⊢ (𝑥 = ∪ 𝐴 → (𝑦<P 𝑥 ↔ 𝑦<P ∪ 𝐴)) | |
21 | 20 | imbi1d 344 | . . . . 5 ⊢ (𝑥 = ∪ 𝐴 → ((𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧) ↔ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
22 | 21 | ralbidv 3199 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → (∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧) ↔ ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
23 | 19, 22 | anbi12d 632 | . . 3 ⊢ (𝑥 = ∪ 𝐴 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))) |
24 | 23 | rspcev 3625 | . 2 ⊢ ((∪ 𝐴 ∈ P ∧ (∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
25 | 1, 16, 24 | syl2anc 586 | 1 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 class class class wbr 5068 Pcnp 10283 <P cltp 10287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-omul 8109 df-er 8291 df-ni 10296 df-mi 10298 df-lti 10299 df-ltpq 10334 df-enq 10335 df-nq 10336 df-ltnq 10342 df-np 10405 df-ltp 10409 |
This theorem is referenced by: supsrlem 10535 |
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