Proof of Theorem prproe
Step | Hyp | Ref
| Expression |
1 | | elpri 4579 |
. . 3
⊢ (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) |
2 | | simprrr 782 |
. . . . . . 7
⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ 𝑉) |
3 | | necom 2996 |
. . . . . . . . . 10
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) |
4 | | neeq2 3006 |
. . . . . . . . . . . 12
⊢ (𝐴 = 𝐶 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶)) |
5 | 4 | eqcoms 2747 |
. . . . . . . . . . 11
⊢ (𝐶 = 𝐴 → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶)) |
6 | 5 | biimpcd 252 |
. . . . . . . . . 10
⊢ (𝐵 ≠ 𝐴 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶)) |
7 | 3, 6 | sylbi 220 |
. . . . . . . . 9
⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶)) |
8 | 7 | adantr 484 |
. . . . . . . 8
⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐴 → 𝐵 ≠ 𝐶)) |
9 | 8 | impcom 411 |
. . . . . . 7
⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ≠ 𝐶) |
10 | | eldifsn 4716 |
. . . . . . 7
⊢ (𝐵 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐵 ∈ 𝑉 ∧ 𝐵 ≠ 𝐶)) |
11 | 2, 9, 10 | sylanbrc 586 |
. . . . . 6
⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ (𝑉 ∖ {𝐶})) |
12 | | eleq1 2827 |
. . . . . . 7
⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵})) |
13 | 12 | adantl 485 |
. . . . . 6
⊢ (((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐵 ∈ {𝐴, 𝐵})) |
14 | | prid2g 4693 |
. . . . . . . . 9
⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵}) |
15 | 14 | adantl 485 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵}) |
16 | 15 | adantl 485 |
. . . . . . 7
⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ {𝐴, 𝐵}) |
17 | 16 | adantl 485 |
. . . . . 6
⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐵 ∈ {𝐴, 𝐵}) |
18 | 11, 13, 17 | rspcedvd 3554 |
. . . . 5
⊢ ((𝐶 = 𝐴 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}) |
19 | 18 | ex 416 |
. . . 4
⊢ (𝐶 = 𝐴 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})) |
20 | | simprrl 781 |
. . . . . . 7
⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ 𝑉) |
21 | | neeq2 3006 |
. . . . . . . . . . 11
⊢ (𝐵 = 𝐶 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)) |
22 | 21 | eqcoms 2747 |
. . . . . . . . . 10
⊢ (𝐶 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶)) |
23 | 22 | biimpcd 252 |
. . . . . . . . 9
⊢ (𝐴 ≠ 𝐵 → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶)) |
24 | 23 | adantr 484 |
. . . . . . . 8
⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐶 = 𝐵 → 𝐴 ≠ 𝐶)) |
25 | 24 | impcom 411 |
. . . . . . 7
⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ≠ 𝐶) |
26 | | eldifsn 4716 |
. . . . . . 7
⊢ (𝐴 ∈ (𝑉 ∖ {𝐶}) ↔ (𝐴 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) |
27 | 20, 25, 26 | sylanbrc 586 |
. . . . . 6
⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ (𝑉 ∖ {𝐶})) |
28 | | eleq1 2827 |
. . . . . . 7
⊢ (𝑣 = 𝐴 → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵})) |
29 | 28 | adantl 485 |
. . . . . 6
⊢ (((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) ∧ 𝑣 = 𝐴) → (𝑣 ∈ {𝐴, 𝐵} ↔ 𝐴 ∈ {𝐴, 𝐵})) |
30 | | prid1g 4692 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) |
31 | 30 | adantr 484 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵}) |
32 | 31 | adantl 485 |
. . . . . . 7
⊢ ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ {𝐴, 𝐵}) |
33 | 32 | adantl 485 |
. . . . . 6
⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → 𝐴 ∈ {𝐴, 𝐵}) |
34 | 27, 29, 33 | rspcedvd 3554 |
. . . . 5
⊢ ((𝐶 = 𝐵 ∧ (𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}) |
35 | 34 | ex 416 |
. . . 4
⊢ (𝐶 = 𝐵 → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})) |
36 | 19, 35 | jaoi 857 |
. . 3
⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})) |
37 | 1, 36 | syl 17 |
. 2
⊢ (𝐶 ∈ {𝐴, 𝐵} → ((𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})) |
38 | 37 | 3impib 1118 |
1
⊢ ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}) |