Theorem plngrnssp 29136

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 4028	. . 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 29134	. . . 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 3929	. 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 29135	. 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 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412   ∖ cdif 3896   class class class wbr 5103  ran crn 5656  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  TarskiGcstrkg 28768  Itvcitv 28774  LineGclng 28775  hpGchpg 29114  hlGcplng 29130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-plng 29131

This theorem is used by:  lnssplng  29149  mirplncl  29152  perpeqlem  29226  perpeq  29227