| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 2wlkdlem7 | Structured version Visualization version GIF version | ||
| Description: Lemma 7 for 2wlkd 29956. (Contributed by AV, 14-Feb-2021.) |
| Ref | Expression |
|---|---|
| 2wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
| 2wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
| 2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
| 2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
| 2wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
| Ref | Expression |
|---|---|
| 2wlkdlem7 | ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
| 2 | 2wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
| 3 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
| 4 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
| 5 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
| 6 | 1, 2, 3, 4, 5 | 2wlkdlem6 29951 | . 2 ⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾))) |
| 7 | elfvex 6944 | . . 3 ⊢ (𝐵 ∈ (𝐼‘𝐽) → 𝐽 ∈ V) | |
| 8 | elfvex 6944 | . . 3 ⊢ (𝐵 ∈ (𝐼‘𝐾) → 𝐾 ∈ V) | |
| 9 | 7, 8 | anim12i 613 | . 2 ⊢ ((𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾)) → (𝐽 ∈ V ∧ 𝐾 ∈ V)) |
| 10 | 6, 9 | syl 17 | 1 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 {cpr 4628 ‘cfv 6561 〈“cs2 14880 〈“cs3 14881 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: 2wlkdlem8 29953 2trld 29958 |
| Copyright terms: Public domain | W3C validator |