Theorem axtgcont 28432

Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 28431. (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)
axtgcont.3	⊢ (𝜑 → 𝐴 ∈ 𝑃)
axtgcont.4	⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) → 𝑢 ∈ (𝐴𝐼𝑣))

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑣,𝑏,𝐴,𝑢,𝑥,𝑦   𝐼,𝑏   𝑣,𝑢,𝑥,𝑦,𝐼   𝑃,𝑏,𝑢,𝑣,𝑥,𝑦   𝑆,𝑏,𝑥   𝑇,𝑏,𝑥,𝑦   − ,𝑏,𝑢,𝑣,𝑥,𝑦   𝜑,𝑢,𝑣   𝑢,𝑆,𝑣   𝑢,𝑇,𝑣   𝑢,𝐴,𝑥,𝑦

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

Proof of Theorem axtgcont

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axtgcont.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
2		axtgcont.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) → 𝑢 ∈ (𝐴𝐼𝑣))
3	2	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
4	3	ralrimivva 3189	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5		simplr 768	. . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6		simpll 766	. . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7		simpr 484	. . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
8	6, 7	oveq12d 7432	. . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
9	5, 8	eleq12d 2827	. . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
10	9	cbvraldva 3226	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) → (∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
11	10	cbvraldva 3226	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
12	11	rspcev 3606	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦))
13	1, 4, 12	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦))
14		axtrkg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
15		axtrkg.d	. . 3 ⊢ − = (dist‘𝐺)
16		axtrkg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
17		axtrkg.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
18		axtgcont.1	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑃)
19		axtgcont.2	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑃)
20	14, 15, 16, 17, 18, 19	axtgcont1 28431	. 2 ⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
21	13, 20	mpd 15	1 ⊢ (𝜑 → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   ⊆ wss 3933  ‘cfv 6542  (class class class)co 7414  Basecbs 17230  distcds 17286  TarskiGcstrkg 28390  Itvcitv 28396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5278  ax-nul 5288

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-iota 6495  df-fv 6550  df-ov 7417  df-trkgb 28412  df-trkg 28416

This theorem is referenced by:  f1otrg  28834