Theorem axtgcont1 26374

Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom (scheme) asserts that any two sets 𝑆 and 𝑇 (of points) such that the elements of 𝑆 precede the elements of 𝑇 with respect to some point 𝑎 (that is, 𝑥 is between 𝑎 and 𝑦 whenever 𝑥 is in 𝑋 and 𝑦 is in 𝑌) are separated by some point 𝑏; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.)

Hypotheses

Ref	Expression
axtrkg.p	⊢ 𝑃 = (Base‘𝐺)
axtrkg.d	⊢ − = (dist‘𝐺)
axtrkg.i	⊢ 𝐼 = (Itv‘𝐺)
axtrkg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgcont.1	⊢ (𝜑 → 𝑆 ⊆ 𝑃)
axtgcont.2	⊢ (𝜑 → 𝑇 ⊆ 𝑃)

Assertion

Ref	Expression
axtgcont1	⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)))

Distinct variable groups:   𝑥,𝑦   𝑎,𝑏,𝑥,𝑦,𝐼   𝑃,𝑎,𝑏,𝑥,𝑦   𝑆,𝑎,𝑏,𝑥   𝑇,𝑎,𝑏,𝑥,𝑦   − ,𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)   𝑆(𝑦)   𝐺(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem axtgcont1

Dummy variables 𝑓 𝑖 𝑝 𝑧 𝑣 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-trkg 26359	. . . . 5 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2		inss1 4135	. . . . . 6 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3		inss2 4136	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
4	2, 3	sstri 3903	. . . . 5 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGB
5	1, 4	eqsstri 3928	. . . 4 ⊢ TarskiG ⊆ TarskiGB
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sseldi 3892	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiGB)
8		axtrkg.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
9		axtrkg.d	. . . . . 6 ⊢ − = (dist‘𝐺)
10		axtrkg.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
11	8, 9, 10	istrkgb 26361	. . . . 5 ⊢ (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
12	11	simprbi 500	. . . 4 ⊢ (𝐺 ∈ TarskiGB → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
13	12	simp3d 1141	. . 3 ⊢ (𝐺 ∈ TarskiGB → ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
14	7, 13	syl 17	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
15		axtgcont.1	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑃)
16	8	fvexi 6677	. . . . . 6 ⊢ 𝑃 ∈ V
17	16	ssex 5195	. . . . 5 ⊢ (𝑆 ⊆ 𝑃 → 𝑆 ∈ V)
18		elpwg 4500	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝑃 ↔ 𝑆 ⊆ 𝑃))
19	15, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → (𝑆 ∈ 𝒫 𝑃 ↔ 𝑆 ⊆ 𝑃))
20	15, 19	mpbird 260	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝑃)
21		axtgcont.2	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑃)
22	16	ssex 5195	. . . . 5 ⊢ (𝑇 ⊆ 𝑃 → 𝑇 ∈ V)
23		elpwg 4500	. . . . 5 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑃 ↔ 𝑇 ⊆ 𝑃))
24	21, 22, 23	3syl 18	. . . 4 ⊢ (𝜑 → (𝑇 ∈ 𝒫 𝑃 ↔ 𝑇 ⊆ 𝑃))
25	21, 24	mpbird 260	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑃)
26		raleq 3323	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
27	26	rexbidv 3221	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
28		raleq 3323	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
29	28	rexbidv 3221	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
30	27, 29	imbi12d 348	. . . 4 ⊢ (𝑠 = 𝑆 → ((∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
31		raleq 3323	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
32	31	rexralbidv 3225	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
33		raleq 3323	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
34	33	rexralbidv 3225	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
35	32, 34	imbi12d 348	. . . 4 ⊢ (𝑡 = 𝑇 → ((∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
36	30, 35	rspc2v 3553	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝑃 ∧ 𝑇 ∈ 𝒫 𝑃) → (∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
37	20, 25, 36	syl2anc 587	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
38	14, 37	mpd 15	1 ⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409  [wsbc 3698   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16554  distcds 16645  TarskiGcstrkg 26336  TarskiG_Ccstrkgc 26337  TarskiG_Bcstrkgb 26338  TarskiG_CBcstrkgcb 26339  Itvcitv 26342  LineGclng 26343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-trkgb 26355  df-trkg 26359

This theorem is referenced by:  axtgcont  26375