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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 10467 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P) | |
2 | ltrelpr 10413 | . . . . . . . . 9 ⊢ <P ⊆ (P × P) | |
3 | 2 | brel 5610 | . . . . . . . 8 ⊢ (𝑦<P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P)) |
4 | 3 | simpld 497 | . . . . . . 7 ⊢ (𝑦<P 𝑥 → 𝑦 ∈ P) |
5 | 4 | ralimi 3159 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ P) |
6 | dfss3 3949 | . . . . . 6 ⊢ (𝐴 ⊆ P ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ P) | |
7 | 5, 6 | sylibr 236 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P) |
8 | 7 | rexlimivw 3281 | . . . 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 10468 | . . . . . 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 3180 | . . . 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 3182 | . . . 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 5062 | . . . . . 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 3196 | . . . 4 ⊢ (𝑥 = ∪ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴<P 𝑦)) |
20 | breq2 5063 | . . . . . 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 3196 | . . . 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 3620 | . 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 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 ⊆ wss 3929 ∅c0 4284 ∪ cuni 4831 class class class wbr 5059 Pcnp 10274 <P cltp 10278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-oadd 8099 df-omul 8100 df-er 8282 df-ni 10287 df-mi 10289 df-lti 10290 df-ltpq 10325 df-enq 10326 df-nq 10327 df-ltnq 10333 df-np 10396 df-ltp 10400 |
This theorem is referenced by: supsrlem 10526 |
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