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Mirrors > Home > MPE Home > Th. List > 1wlkdlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 1wlkd 29432. (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 4662 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑋}) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ {𝑋}) |
5 | 1wlkd.l | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
6 | 4, 5 | eleqtrrd 2836 | . 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 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑉) |
10 | prssg 4822 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽))) | |
11 | 1, 9, 10 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽))) |
12 | 7, 11 | mpbird 256 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽))) |
13 | 12 | simpld 495 | . 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3948 {csn 4628 {cpr 4630 ‘cfv 6543 ⟨“cs1 14547 ⟨“cs2 14794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 |
This theorem is referenced by: 1wlkdlem3 29430 1wlkdlem4 29431 |
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