suprzclex - Intuitionistic Logic Explorer

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Theorem suprzclex 9441

Description: The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)

**Hypotheses**
Ref	Expression
suprzclex.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
suprzclex.ss	⊢ (𝜑 → 𝐴 ⊆ ℤ)

**Assertion**
Ref	Expression
suprzclex	⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)

Distinct variable groups: 𝑥,𝐴,𝑦,𝑧 𝜑,𝑥,𝑧 Allowed substitution hint: 𝜑(𝑦)

Proof of Theorem suprzclex Dummy variables 𝑤 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lttri3 8123	. . . . . 6 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		suprzclex.ex	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
4	2, 3	supclti 7073	. . . 4 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5	4	ltm1d 8976	. . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 1) < sup(𝐴, ℝ, < ))
6		suprzclex.ss	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
7		zssre 9350	. . . . 5 ⊢ ℤ ⊆ ℝ
8	6, 7	sstrdi 3196	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9		peano2rem 8310	. . . . 5 ⊢ (sup(𝐴, ℝ, < ) ∈ ℝ → (sup(𝐴, ℝ, < ) − 1) ∈ ℝ)
10	4, 9	syl 14	. . . 4 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 1) ∈ ℝ)
11	3, 8, 10	suprlubex 8996	. . 3 ⊢ (𝜑 → ((sup(𝐴, ℝ, < ) − 1) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧))
12	5, 11	mpbid 147	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 1) < 𝑧)
13	6	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝐴 ⊆ ℤ)
14	13	sselda 3184	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
15	7, 14	sselid 3182	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
16	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ∈ ℝ)
17	16	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℝ)
18		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ 𝐴)
19	13, 18	sseldd 3185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ ℤ)
20		zre 9347	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
21	19, 20	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ∈ ℝ)
22		peano2re 8179	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → (𝑧 + 1) ∈ ℝ)
23	21, 22	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (𝑧 + 1) ∈ ℝ)
24	23	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → (𝑧 + 1) ∈ ℝ)
25	3	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
26	8	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝐴 ⊆ ℝ)
27		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
28	25, 26, 27	suprubex 8995	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ sup(𝐴, ℝ, < ))
29		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) − 1) < 𝑧)
30		1red 8058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 1 ∈ ℝ)
31	16, 30, 21	ltsubaddd 8585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → ((sup(𝐴, ℝ, < ) − 1) < 𝑧 ↔ sup(𝐴, ℝ, < ) < (𝑧 + 1)))
32	29, 31	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) < (𝑧 + 1))
33	32	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → sup(𝐴, ℝ, < ) < (𝑧 + 1))
34	15, 17, 24, 28, 33	lelttrd 8168	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 < (𝑧 + 1))
35	19	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑧 ∈ ℤ)
36		zleltp1 9398	. . . . . . . 8 ⊢ ((𝑤 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑤 ≤ 𝑧 ↔ 𝑤 < (𝑧 + 1)))
37	14, 35, 36	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤 < (𝑧 + 1)))
38	34, 37	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ≤ 𝑧)
39	38	ralrimiva 2570	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧)
40		breq2 4038	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑤))
41	40	cbvrexv 2730	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑤 ∈ 𝐴 𝑦 < 𝑤)
42	41	imbi2i 226	. . . . . . . . . . 11 ⊢ ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤))
43	42	ralbii 2503	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤))
44	43	anbi2i 457	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤)))
45	44	rexbii 2504	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤)))
46	3, 45	sylib 122	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤)))
47	46	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐴 𝑦 < 𝑤)))
48	13, 7	sstrdi 3196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝐴 ⊆ ℝ)
49	47, 48, 21	suprleubex 8998	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) ≤ 𝑧 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝑧))
50	39, 49	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ≤ 𝑧)
51	47, 48, 18	suprubex 8995	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → 𝑧 ≤ sup(𝐴, ℝ, < ))
52	16, 21	letri3d 8159	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → (sup(𝐴, ℝ, < ) = 𝑧 ↔ (sup(𝐴, ℝ, < ) ≤ 𝑧 ∧ 𝑧 ≤ sup(𝐴, ℝ, < ))))
53	50, 51, 52	mpbir2and 946	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) = 𝑧)
54	53, 18	eqeltrd 2273	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ (sup(𝐴, ℝ, < ) − 1) < 𝑧)) → sup(𝐴, ℝ, < ) ∈ 𝐴)
55	12, 54	rexlimddv 2619	1 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2167 ∀wral 2475 ∃wrex 2476 ⊆ wss 3157 class class class wbr 4034 (class class class)co 5925 supcsup 7057 ℝcr 7895 1c1 7897 + caddc 7899 < clt 8078 ≤ cle 8079 − cmin 8214 ℤcz 9343

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-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 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-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-br 4035 df-opab 4096 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-sup 7059 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344

This theorem is referenced by: infssuzcldc 10342 gcddvds 12155

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