Proof of Theorem bnj1417
| Step | Hyp | Ref
| Expression |
| 1 | | bnj1417.1 |
. . . 4
⊢ (𝜑 ↔ 𝑅 FrSe 𝐴) |
| 2 | 1 | biimpi 216 |
. . 3
⊢ (𝜑 → 𝑅 FrSe 𝐴) |
| 3 | | bnj1417.4 |
. . . . . 6
⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)) |
| 4 | | bnj1418 35054 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥) |
| 5 | 4 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥) |
| 6 | 3, 2 | bnj835 34773 |
. . . . . . . . . . . 12
⊢ (𝜃 → 𝑅 FrSe 𝐴) |
| 7 | | df-bnj15 34707 |
. . . . . . . . . . . . 13
⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) |
| 8 | 7 | simplbi 497 |
. . . . . . . . . . . 12
⊢ (𝑅 FrSe 𝐴 → 𝑅 Fr 𝐴) |
| 9 | 6, 8 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜃 → 𝑅 Fr 𝐴) |
| 10 | | bnj213 34896 |
. . . . . . . . . . . 12
⊢
pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴 |
| 11 | 10 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴) |
| 12 | | frirr 5661 |
. . . . . . . . . . 11
⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
| 13 | 9, 11, 12 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥) |
| 14 | 5, 13 | pm2.65da 817 |
. . . . . . . . 9
⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) |
| 15 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦𝜑 |
| 16 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦 𝑥 ∈ 𝐴 |
| 17 | | bnj1417.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓)) |
| 18 | 17 | bnj1095 34795 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → ∀𝑦𝜒) |
| 19 | 18 | nf5i 2146 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦𝜒 |
| 20 | 15, 16, 19 | nf3an 1901 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) |
| 21 | 3, 20 | nfxfr 1853 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦𝜃 |
| 22 | 6 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴) |
| 23 | | simplr 769 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) |
| 24 | 10, 23 | sselid 3981 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴) |
| 25 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 26 | | bnj1125 35006 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅)) |
| 27 | 22, 24, 25, 26 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅)) |
| 28 | | bnj1147 35008 |
. . . . . . . . . . . . . . . . . 18
⊢
trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
| 29 | 28, 25 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴) |
| 30 | | bnj906 34944 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
| 31 | 22, 29, 30 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
| 32 | 31, 23 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)) |
| 33 | 27, 32 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 34 | 17 | biimpi 216 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓)) |
| 35 | 3, 34 | bnj837 34775 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓)) |
| 36 | 35 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓)) |
| 37 | | bnj1418 35054 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥) |
| 38 | 37 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥) |
| 39 | | rsp 3247 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑦 ∈
𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓) → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))) |
| 40 | 36, 24, 38, 39 | syl3c 66 |
. . . . . . . . . . . . . . 15
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → [𝑦 / 𝑥]𝜓) |
| 41 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑦 ∈ V |
| 42 | | bnj1417.2 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
| 43 | | eleq1w 2824 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))) |
| 44 | | bnj1318 35039 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅)) |
| 45 | 44 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))) |
| 46 | 43, 45 | bitrd 279 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))) |
| 47 | 46 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))) |
| 48 | 42, 47 | bitrid 283 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))) |
| 49 | 41, 48 | sbcie 3830 |
. . . . . . . . . . . . . . 15
⊢
([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 50 | 40, 49 | sylib 218 |
. . . . . . . . . . . . . 14
⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 51 | 33, 50 | pm2.65da 817 |
. . . . . . . . . . . . 13
⊢ ((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 52 | 51 | ex 412 |
. . . . . . . . . . . 12
⊢ (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))) |
| 53 | 21, 52 | ralrimi 3257 |
. . . . . . . . . . 11
⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 54 | | ralnex 3072 |
. . . . . . . . . . 11
⊢
(∀𝑦 ∈
pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅) ↔ ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 55 | 53, 54 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 56 | | eliun 4995 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) |
| 57 | 55, 56 | sylnibr 329 |
. . . . . . . . 9
⊢ (𝜃 → ¬ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) |
| 58 | | ioran 986 |
. . . . . . . . 9
⊢ (¬
(𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ↔ (¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∧ ¬ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) |
| 59 | 14, 57, 58 | sylanbrc 583 |
. . . . . . . 8
⊢ (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) |
| 60 | 3 | simp2bi 1147 |
. . . . . . . . . . 11
⊢ (𝜃 → 𝑥 ∈ 𝐴) |
| 61 | | bnj1417.5 |
. . . . . . . . . . . 12
⊢ 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) |
| 62 | 61 | bnj1414 35051 |
. . . . . . . . . . 11
⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵) |
| 63 | 6, 60, 62 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵) |
| 64 | 63 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵)) |
| 65 | 61 | bnj1138 34802 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))) |
| 66 | 64, 65 | bitrdi 287 |
. . . . . . . 8
⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪
𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))) |
| 67 | 59, 66 | mtbird 325 |
. . . . . . 7
⊢ (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
| 68 | 67, 42 | sylibr 234 |
. . . . . 6
⊢ (𝜃 → 𝜓) |
| 69 | 3, 68 | sylbir 235 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝜓) |
| 70 | 69 | 3exp 1120 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓))) |
| 71 | 70 | ralrimiv 3145 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) |
| 72 | 17 | bnj1204 35026 |
. . 3
⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) → ∀𝑥 ∈ 𝐴 𝜓) |
| 73 | 2, 71, 72 | syl2anc 584 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
| 74 | 42 | ralbii 3093 |
. 2
⊢
(∀𝑥 ∈
𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
| 75 | 73, 74 | sylib 218 |
1
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |