Theorem suplocexpr 8040

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 4112	. . . . . 6 ⊢ (𝑎 = 𝑤 → (𝑎 <Q 𝑢 ↔ 𝑤 <Q 𝑢))
5	4	cbvrexv 2779	. . . . 5 ⊢ (∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢)
6	5	rabbii 2800	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}
7	6	opeq2i 3887	. . 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 8037	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
9	1, 2, 3, 7	suplocexprlemub 8038	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
10	1, 2, 3, 7	suplocexprlemlub 8039	. . 3 ⊢ (𝜑 → (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
11	10	ralrimivw 2616	. 2 ⊢ (𝜑 → ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
12		breq1 4112	. . . . . 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 673	. . . . 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 2542	. . . 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 4113	. . . . . 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 2542	. . . 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 2921	. 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 1272	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 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524  ⟨cop 3692  ∪ cuni 3914  ∩ cint 3949   class class class wbr 4109   “ cima 4752  1^st c1st 6332  2^nd c2nd 6333  Qcnq 7595   <_Q cltq 7600  Pcnp 7606  <_P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  suplocsrlempr  8122