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Mirrors > Home > MPE Home > Th. List > tglnssp | Structured version Visualization version GIF version |
Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
tglngval.z | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Ref | Expression |
---|---|
tglnssp | ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tglngval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
6 | tglngval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
7 | tglngval.z | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tglngval 25446 | . 2 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) |
9 | ssrab2 3687 | . 2 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃 | |
10 | 8, 9 | syl6eqss 3655 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-trkg 25352 |
This theorem is referenced by: tglineelsb2 25527 tglinecom 25530 |
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