Proof of Theorem sprsymrelfolem2
Step | Hyp | Ref
| Expression |
1 | | df-br 5075 |
. . . . . . . 8
⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
2 | | simpl 483 |
. . . . . . . . . 10
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → 𝑉 ∈ 𝑊) |
3 | | ssel 3914 |
. . . . . . . . . . . . 13
⊢ (𝑅 ⊆ (𝑉 × 𝑉) → (〈𝑥, 𝑦〉 ∈ 𝑅 → 〈𝑥, 𝑦〉 ∈ (𝑉 × 𝑉))) |
4 | 3 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (〈𝑥, 𝑦〉 ∈ 𝑅 → 〈𝑥, 𝑦〉 ∈ (𝑉 × 𝑉))) |
5 | 4 | imp 407 |
. . . . . . . . . . 11
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → 〈𝑥, 𝑦〉 ∈ (𝑉 × 𝑉)) |
6 | | opelxp 5625 |
. . . . . . . . . . 11
⊢
(〈𝑥, 𝑦〉 ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
7 | 5, 6 | sylib 217 |
. . . . . . . . . 10
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
8 | | prelspr 44938 |
. . . . . . . . . 10
⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → {𝑥, 𝑦} ∈ (Pairs‘𝑉)) |
9 | 2, 7, 8 | syl2an2r 682 |
. . . . . . . . 9
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ 〈𝑥, 𝑦〉 ∈ 𝑅) → {𝑥, 𝑦} ∈ (Pairs‘𝑉)) |
10 | 9 | ex 413 |
. . . . . . . 8
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (〈𝑥, 𝑦〉 ∈ 𝑅 → {𝑥, 𝑦} ∈ (Pairs‘𝑉))) |
11 | 1, 10 | syl5bi 241 |
. . . . . . 7
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉))) |
12 | 11 | 3adant3 1131 |
. . . . . 6
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉))) |
13 | 12 | imp 407 |
. . . . 5
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ (Pairs‘𝑉)) |
14 | | vex 3436 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
15 | | vex 3436 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
16 | | vex 3436 |
. . . . . . . 8
⊢ 𝑎 ∈ V |
17 | | vex 3436 |
. . . . . . . 8
⊢ 𝑏 ∈ V |
18 | 14, 15, 16, 17 | preq12b 4781 |
. . . . . . 7
⊢ ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎))) |
19 | | breq12 5079 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑅𝑦 ↔ 𝑎𝑅𝑏)) |
20 | 19 | biimpd 228 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑅𝑦 → 𝑎𝑅𝑏)) |
21 | 20 | com12 32 |
. . . . . . . . . . . 12
⊢ (𝑥𝑅𝑦 → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏)) |
22 | 21 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏)) |
23 | 22 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏)) |
24 | 23 | com12 32 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏)) |
25 | | rsp2 3138 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑥 ∈
𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
26 | 25 | ancomsd 466 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
27 | 26 | imp 407 |
. . . . . . . . . . . . . . . . 17
⊢
((∀𝑥 ∈
𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) |
28 | 27 | biimpd 228 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑥 ∈
𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
29 | 28 | ex 413 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
30 | 29 | 3ad2ant3 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
31 | 30 | com23 86 |
. . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥))) |
32 | 31 | imp 407 |
. . . . . . . . . . . 12
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥)) |
33 | 32 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥)) |
34 | | eleq1 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝑉 ↔ 𝑎 ∈ 𝑉)) |
35 | | eleq1 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑏 → (𝑥 ∈ 𝑉 ↔ 𝑏 ∈ 𝑉)) |
36 | 34, 35 | bi2anan9r 637 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))) |
37 | | breq12 5079 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 = 𝑎 ∧ 𝑥 = 𝑏) → (𝑦𝑅𝑥 ↔ 𝑎𝑅𝑏)) |
38 | 37 | ancoms 459 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → (𝑦𝑅𝑥 ↔ 𝑎𝑅𝑏)) |
39 | 36, 38 | imbi12d 345 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → (((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏))) |
40 | 39 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → (((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏))) |
41 | 33, 40 | mpbid 231 |
. . . . . . . . . 10
⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏)) |
42 | 41 | expimpd 454 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏)) |
43 | 24, 42 | jaoi 854 |
. . . . . . . 8
⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏)) |
44 | 43 | com12 32 |
. . . . . . 7
⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → 𝑎𝑅𝑏)) |
45 | 18, 44 | syl5bi 241 |
. . . . . 6
⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)) |
46 | 45 | ralrimivva 3123 |
. . . . 5
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)) |
47 | | sprsymrelfo.q |
. . . . . . 7
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} |
48 | 47 | eleq2i 2830 |
. . . . . 6
⊢ ({𝑥, 𝑦} ∈ 𝑄 ↔ {𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}) |
49 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑞 = {𝑥, 𝑦} → (𝑞 = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = {𝑎, 𝑏})) |
50 | 49 | imbi1d 342 |
. . . . . . . 8
⊢ (𝑞 = {𝑥, 𝑦} → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
51 | 50 | 2ralbidv 3129 |
. . . . . . 7
⊢ (𝑞 = {𝑥, 𝑦} → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
52 | 51 | elrab 3624 |
. . . . . 6
⊢ ({𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
53 | 48, 52 | bitri 274 |
. . . . 5
⊢ ({𝑥, 𝑦} ∈ 𝑄 ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
54 | 13, 46, 53 | sylanbrc 583 |
. . . 4
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ 𝑄) |
55 | | eqidd 2739 |
. . . 4
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} = {𝑥, 𝑦}) |
56 | | eqeq1 2742 |
. . . . 5
⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 = {𝑥, 𝑦} ↔ {𝑥, 𝑦} = {𝑥, 𝑦})) |
57 | 56 | rspcev 3561 |
. . . 4
⊢ (({𝑥, 𝑦} ∈ 𝑄 ∧ {𝑥, 𝑦} = {𝑥, 𝑦}) → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}) |
58 | 54, 55, 57 | syl2anc 584 |
. . 3
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}) |
59 | 58 | ex 413 |
. 2
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})) |
60 | 47 | eleq2i 2830 |
. . . . . 6
⊢ (𝑐 ∈ 𝑄 ↔ 𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}) |
61 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑞 = 𝑐 → (𝑞 = {𝑎, 𝑏} ↔ 𝑐 = {𝑎, 𝑏})) |
62 | 61 | imbi1d 342 |
. . . . . . . 8
⊢ (𝑞 = 𝑐 → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
63 | 62 | 2ralbidv 3129 |
. . . . . . 7
⊢ (𝑞 = 𝑐 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
64 | 63 | elrab 3624 |
. . . . . 6
⊢ (𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
65 | 60, 64 | bitri 274 |
. . . . 5
⊢ (𝑐 ∈ 𝑄 ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏))) |
66 | | eleq1 2826 |
. . . . . . . . . . 11
⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) ↔ {𝑥, 𝑦} ∈ (Pairs‘𝑉))) |
67 | | prsprel 44939 |
. . . . . . . . . . . 12
⊢ (({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
68 | 14, 15, 67 | mpanr12 702 |
. . . . . . . . . . 11
⊢ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
69 | 66, 68 | syl6bi 252 |
. . . . . . . . . 10
⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
70 | 69 | com12 32 |
. . . . . . . . 9
⊢ (𝑐 ∈ (Pairs‘𝑉) → (𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
71 | 70 | adantr 481 |
. . . . . . . 8
⊢ ((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) → (𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
72 | 71 | imp 407 |
. . . . . . 7
⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
73 | | preq1 4669 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑥 → {𝑎, 𝑏} = {𝑥, 𝑏}) |
74 | 73 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑥, 𝑏})) |
75 | | breq1 5077 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (𝑎𝑅𝑏 ↔ 𝑥𝑅𝑏)) |
76 | 74, 75 | imbi12d 345 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑥 → ((𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏))) |
77 | | preq2 4670 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑦 → {𝑥, 𝑏} = {𝑥, 𝑦}) |
78 | 77 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑦 → (𝑐 = {𝑥, 𝑏} ↔ 𝑐 = {𝑥, 𝑦})) |
79 | | breq2 5078 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑦 → (𝑥𝑅𝑏 ↔ 𝑥𝑅𝑦)) |
80 | 78, 79 | imbi12d 345 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑦 → ((𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))) |
81 | 76, 80 | rspc2v 3570 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))) |
82 | 81 | a1d 25 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑐 ∈ (Pairs‘𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))) |
83 | 82 | imp4c 424 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦)) |
84 | 72, 83 | mpcom 38 |
. . . . . 6
⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦) |
85 | 84 | a1d 25 |
. . . . 5
⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦)) |
86 | 65, 85 | sylanb 581 |
. . . 4
⊢ ((𝑐 ∈ 𝑄 ∧ 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦)) |
87 | 86 | rexlimiva 3210 |
. . 3
⊢
(∃𝑐 ∈
𝑄 𝑐 = {𝑥, 𝑦} → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦)) |
88 | 87 | com12 32 |
. 2
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)) |
89 | 59, 88 | impbid 211 |
1
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})) |