Theorem suplocexpr 7809

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 4037	. . . . . 6 ⊢ (𝑎 = 𝑤 → (𝑎 <Q 𝑢 ↔ 𝑤 <Q 𝑢))
5	4	cbvrexv 2730	. . . . 5 ⊢ (∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢)
6	5	rabbii 2749	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}
7	6	opeq2i 3813	. . 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 7806	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
9	1, 2, 3, 7	suplocexprlemub 7807	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
10	1, 2, 3, 7	suplocexprlemlub 7808	. . 3 ⊢ (𝜑 → (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
11	10	ralrimivw 2571	. 2 ⊢ (𝜑 → ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
12		breq1 4037	. . . . . 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 668	. . . . 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 2497	. . . 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 4038	. . . . . 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 2497	. . . 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 473	. . 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 2868	. 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 1247	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 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  ⟨cop 3626  ∪ cuni 3840  ∩ cint 3875   class class class wbr 4034   “ cima 4667  1^st c1st 6205  2^nd c2nd 6206  Qcnq 7364   <_Q cltq 7369  Pcnp 7375  <_P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iltp 7554

This theorem is referenced by:  suplocsrlempr  7891