Theorem plngrnssp 28983

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 4033	. . 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 740	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ 𝐻)
4		simpr 488	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
5	3, 4	eleqtrd 2864	. . . 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 740	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐺 ∈ TarskiG)
12		simpllr 785	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑎 ∈ ran 𝐿)
13		simplr 778	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
14	6, 7, 8, 9, 11, 12, 13	plngval 28981	. . . 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 2864	. . 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 3934	. 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 28982	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
19	16, 18	r19.29vva 3222	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1097   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  {crab 3414   ∖ cdif 3901   class class class wbr 5100  ran crn 5648  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  TarskiGcstrkg 28593  Itvcitv 28599  LineGclng 28600  hpGchpg 28927  hlGcplng 28977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-plng 28978

This theorem is referenced by:  lnssplng  28996