Theorem suplocexpr 7557

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 3940	. . . . . 6 ⊢ (𝑎 = 𝑤 → (𝑎 <Q 𝑢 ↔ 𝑤 <Q 𝑢))
5	4	cbvrexv 2658	. . . . 5 ⊢ (∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢)
6	5	rabbii 2675	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}
7	6	opeq2i 3717	. . 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 7554	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
9	1, 2, 3, 7	suplocexprlemub 7555	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
10	1, 2, 3, 7	suplocexprlemlub 7556	. . 3 ⊢ (𝜑 → (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
11	10	ralrimivw 2509	. 2 ⊢ (𝜑 → ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
12		breq1 3940	. . . . . 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 657	. . . . 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 2438	. . . 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 3941	. . . . . 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 230	. . . . 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 2438	. . . 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 465	. . 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 2793	. 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 1215	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 103   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {crab 2421  ⟨cop 3535  ∪ cuni 3744  ∩ cint 3779   class class class wbr 3937   “ cima 4550  1^st c1st 6044  2^nd c2nd 6045  Qcnq 7112   <_Q cltq 7117  Pcnp 7123  <_P cltp 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iltp 7302

This theorem is referenced by:  suplocsrlempr  7639