Theorem suplocexpr 8082

Description: An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))

Assertion

Ref	Expression
suplocexpr	⊢ (𝜑 → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Distinct variable groups:   𝑦,𝐴,𝑧,𝑥   𝜑,𝑦,𝑧,𝑥

Proof of Theorem suplocexpr

Dummy variables 𝑎 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		suplocexpr.ub	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
3		suplocexpr.loc	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
4		breq1 4128	. . . . . 6 ⊢ (𝑎 = 𝑤 → (𝑎 <Q 𝑢 ↔ 𝑤 <Q 𝑢))
5	4	cbvrexv 2787	. . . . 5 ⊢ (∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢)
6	5	rabbii 2808	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}
7	6	opeq2i 3903	. . 3 ⊢ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
8	1, 2, 3, 7	suplocexprlemex 8079	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
9	1, 2, 3, 7	suplocexprlemub 8080	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
10	1, 2, 3, 7	suplocexprlemlub 8081	. . 3 ⊢ (𝜑 → (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
11	10	ralrimivw 2624	. 2 ⊢ (𝜑 → ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
12		breq1 4128	. . . . . 6 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
13	12	notbid 677	. . . . 5 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → (¬ 𝑥<P 𝑦 ↔ ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
14	13	ralbidv 2550	. . . 4 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
15		breq2 4129	. . . . . 6 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → (𝑦<P 𝑥 ↔ 𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩))
16	15	imbi1d 231	. . . . 5 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ((𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧) ↔ (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
17	16	ralbidv 2550	. . . 4 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧) ↔ ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
18	14, 17	anbi12d 477	. . 3 ⊢ (𝑥 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))))
19	18	rspcev 2929	. 2 ⊢ ((⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P ∧ (∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
20	8, 9, 11, 19	syl12anc 1276	1 ⊢ (𝜑 → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3708  ∪ cuni 3930  ∩ cint 3965   class class class wbr 4125   “ cima 4772  1^st c1st 6362  2^nd c2nd 6363  Qcnq 7637   <_Q cltq 7642  Pcnp 7648  <_P cltp 7652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iltp 7827

This theorem is referenced by:  suplocsrlempr  8164