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| Mirrors > Home > MPE Home > Th. List > 1wlkdlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for 1wlkd 30160. (Contributed by AV, 22-Jan-2021.) |
| Ref | Expression |
|---|---|
| 1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
| 1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
| 1wlkd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 1wlkd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 1wlkd.l | ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) |
| 1wlkd.j | ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) |
| Ref | Expression |
|---|---|
| 1wlkdlem2 | ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1wlkd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 2 | snidg 4660 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑋}) |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ {𝑋}) |
| 5 | 1wlkd.l | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
| 6 | 4, 5 | eleqtrrd 2844 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝐼‘𝐽)) |
| 7 | 1wlkd.j | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) | |
| 8 | 1wlkd.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑉) |
| 10 | prssg 4819 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽))) | |
| 11 | 1, 9, 10 | syl2an2r 685 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽))) |
| 12 | 7, 11 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽))) |
| 13 | 12 | simpld 494 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ (𝐼‘𝐽)) |
| 14 | 6, 13 | pm2.61dane 3029 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 〈“cs1 14633 〈“cs2 14880 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: 1wlkdlem3 30158 1wlkdlem4 30159 |
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