Theorem plngrnssp 29264

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 4004	. . 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 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑋 ∈ 𝐻)
4		simpr 490	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐻 = (𝑎𝐸𝑟))
5	3, 4	eleqtrd 2862	. . . 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 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝐺 ∈ TarskiG)
12		simpllr 788	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑎 ∈ ran 𝐿)
13		simplr 781	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ 𝑟 ∈ (𝑃 ∖ 𝑎)) ∧ 𝐻 = (𝑎𝐸𝑟)) → 𝑟 ∈ (𝑃 ∖ 𝑎))
14	6, 7, 8, 9, 11, 12, 13	plngval 29262	. . . 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 2862	. . 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 3907	. 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 29263	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
19	16, 18	r19.29vva 3222	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412   ∖ cdif 4044   class class class wbr 5102  ran crn 5651  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  TarskiGcstrkg 28896  Itvcitv 28902  LineGclng 28903  hpGchpg 29242  hlGcplng 29258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-plng 29259

This theorem is used by:  lnssplng  29277  mirplncl  29280  perpeqlem  29354  perpeq  29355