Theorem suplocexpr 7533

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 3932	. . . . . 6 ⊢ (𝑎 = 𝑤 → (𝑎 <Q 𝑢 ↔ 𝑤 <Q 𝑢))
5	4	cbvrexv 2655	. . . . 5 ⊢ (∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢)
6	5	rabbii 2672	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}
7	6	opeq2i 3709	. . 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 7530	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
9	1, 2, 3, 7	suplocexprlemub 7531	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
10	1, 2, 3, 7	suplocexprlemlub 7532	. . 3 ⊢ (𝜑 → (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
11	10	ralrimivw 2506	. 2 ⊢ (𝜑 → ∀𝑦 ∈ P (𝑦<P ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑎 ∈ ∩ (2nd “ 𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
12		breq1 3932	. . . . . 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 656	. . . . 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 2437	. . . 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 3933	. . . . . 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 2437	. . . 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 464	. . 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 2789	. 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 1214	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 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530  ∪ cuni 3736  ∩ cint 3771   class class class wbr 3929   “ cima 4542  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocsrlempr  7615