Theorem plngrnssp 29018

Description: Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)

Hypotheses

Ref	Expression
plngval.p	⊢ 𝑃 = (Base‘𝐺)
plngval.i	⊢ 𝐼 = (Itv‘𝐺)
plngval.1	⊢ 𝐿 = (LineG‘𝐺)
plngval.e	⊢ 𝐸 = (hlG‘𝐺)
plngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
plngrnssp.h	⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
plngrnssp.x	⊢ (𝜑 → 𝑋 ∈ 𝐻)

Assertion

Ref	Expression
plngrnssp	⊢ (𝜑 → 𝑋 ∈ 𝑃)

Proof of Theorem plngrnssp

Dummy variables 𝑡 𝑥 𝑎 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4042	. . 3 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))} ⊆ 𝑃
2		plngrnssp.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐻)
3	2	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ 𝐻)
4		simpr 489	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
5	3, 4	eleqtrd 2871	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ (𝑎𝐸𝑟))
6		plngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
7		plngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
8		plngval.1	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
9		plngval.e	. . . . 5 ⊢ 𝐸 = (hlG‘𝐺)
10		plngval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐺 ∈ TarskiG)
12		simpllr 787	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑎 ∈ ran 𝐿)
13		simplr 780	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
14	6, 7, 8, 9, 11, 12, 13	plngval 29016	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → (𝑎𝐸𝑟) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
15	5, 14	eleqtrd 2871	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥𝐼𝑟))})
16	1, 15	sselid 3943	. 2 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ 𝑃)
17		plngrnssp.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)
18	6, 7, 8, 9, 10, 17	isplng 29017	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
19	16, 18	r19.29vva 3231	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423   ∖ cdif 3910   class class class wbr 5113  ran crn 5663  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  TarskiGcstrkg 28661  Itvcitv 28667  LineGclng 28668  hpGchpg 28997  hlGcplng 29012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-plng 29013

This theorem is referenced by:  lnssplng  29031  mirplncl  29034  perpeqlem  29104  perpeq  29105