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Mirrors > Home > MPE Home > Th. List > 1wlkdlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 1wlkd 30170. (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 4665 | . . . . 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 2842 | . 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 4824 | . . . . 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 3027 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 {csn 4631 {cpr 4633 ‘cfv 6563 〈“cs1 14630 〈“cs2 14877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 |
This theorem is referenced by: 1wlkdlem3 30168 1wlkdlem4 30169 |
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