Theorem axtgcont 27119

Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 27118. (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 1119	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
4	3	ralrimivva 3193	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5		simplr 766	. . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6		simpll 764	. . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7		simpr 485	. . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
8	6, 7	oveq12d 7355	. . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
9	5, 8	eleq12d 2831	. . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
10	9	cbvraldva 3322	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) → (∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
11	10	cbvraldva 3322	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
12	11	rspcev 3570	. . 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 27118	. 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 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070   ⊆ wss 3898  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  distcds 17068  TarskiGcstrkg 27077  Itvcitv 27083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-ov 7340  df-trkgb 27099  df-trkg 27103

This theorem is referenced by:  f1otrg  27521