| Step | Hyp | Ref
| Expression |
| 1 | | tglng.p |
. . . . . . . 8
⊢ 𝑃 = (Base‘𝐺) |
| 2 | | tglng.l |
. . . . . . . 8
⊢ 𝐿 = (LineG‘𝐺) |
| 3 | | tglng.i |
. . . . . . . 8
⊢ 𝐼 = (Itv‘𝐺) |
| 4 | 1, 2, 3 | tglng 28554 |
. . . . . . 7
⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) |
| 5 | 4 | rneqd 5949 |
. . . . . 6
⊢ (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) |
| 6 | 5 | eleq2d 2827 |
. . . . 5
⊢ (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿 ↔ 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))) |
| 7 | 6 | biimpa 476 |
. . . 4
⊢ ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) |
| 8 | | eqid 2737 |
. . . . . 6
⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) |
| 9 | 1 | fvexi 6920 |
. . . . . . 7
⊢ 𝑃 ∈ V |
| 10 | 9 | rabex 5339 |
. . . . . 6
⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V |
| 11 | 8, 10 | elrnmpo 7569 |
. . . . 5
⊢ (𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) |
| 12 | | ssrab2 4080 |
. . . . . . . 8
⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃 |
| 13 | | sseq1 4009 |
. . . . . . . 8
⊢ (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝 ⊆ 𝑃 ↔ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃)) |
| 14 | 12, 13 | mpbiri 258 |
. . . . . . 7
⊢ (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃) |
| 15 | 14 | rexlimivw 3151 |
. . . . . 6
⊢
(∃𝑦 ∈
(𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃) |
| 16 | 15 | rexlimivw 3151 |
. . . . 5
⊢
(∃𝑥 ∈
𝑃 ∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃) |
| 17 | 11, 16 | sylbi 217 |
. . . 4
⊢ (𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑝 ⊆ 𝑃) |
| 18 | 7, 17 | syl 17 |
. . 3
⊢ ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ⊆ 𝑃) |
| 19 | 18 | ralrimiva 3146 |
. 2
⊢ (𝐺 ∈ TarskiG →
∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃) |
| 20 | | unissb 4939 |
. 2
⊢ (∪ ran 𝐿 ⊆ 𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃) |
| 21 | 19, 20 | sylibr 234 |
1
⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃) |