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Theorem tglnssp 27543
Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Baseβ€˜πΊ)
tglngval.l 𝐿 = (LineGβ€˜πΊ)
tglngval.i 𝐼 = (Itvβ€˜πΊ)
tglngval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglngval.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
tglngval.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
tglngval.z (πœ‘ β†’ 𝑋 β‰  π‘Œ)
Assertion
Ref Expression
tglnssp (πœ‘ β†’ (π‘‹πΏπ‘Œ) βŠ† 𝑃)

Proof of Theorem tglnssp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . 3 𝑃 = (Baseβ€˜πΊ)
2 tglngval.l . . 3 𝐿 = (LineGβ€˜πΊ)
3 tglngval.i . . 3 𝐼 = (Itvβ€˜πΊ)
4 tglngval.g . . 3 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 tglngval.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑃)
6 tglngval.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝑃)
7 tglngval.z . . 3 (πœ‘ β†’ 𝑋 β‰  π‘Œ)
81, 2, 3, 4, 5, 6, 7tglngval 27542 . 2 (πœ‘ β†’ (π‘‹πΏπ‘Œ) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
9 ssrab2 4041 . 2 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))} βŠ† 𝑃
108, 9eqsstrdi 4002 1 (πœ‘ β†’ (π‘‹πΏπ‘Œ) βŠ† 𝑃)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  {crab 3406   βŠ† wss 3914  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  TarskiGcstrkg 27418  Itvcitv 27424  LineGclng 27425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-trkg 27444
This theorem is referenced by:  tglineelsb2  27623  tglinecom  27626
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