| Step | Hyp | Ref
| Expression |
| 1 | | acsfiindd.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) |
| 2 | 1 | acsmred 17699 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
| 3 | 2 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝐴 ∈ (Moore‘𝑋)) |
| 4 | | acsfiindd.2 |
. . . . 5
⊢ 𝑁 = (mrCls‘𝐴) |
| 5 | | acsfiindd.3 |
. . . . 5
⊢ 𝐼 = (mrInd‘𝐴) |
| 6 | | simplr 769 |
. . . . 5
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑆 ∈ 𝐼) |
| 7 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) |
| 8 | 7 | elin1d 4204 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑠 ∈ 𝒫 𝑆) |
| 9 | 8 | elpwid 4609 |
. . . . 5
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑠 ⊆ 𝑆) |
| 10 | 3, 4, 5, 6, 9 | mrissmrid 17684 |
. . . 4
⊢ (((𝜑 ∧ 𝑆 ∈ 𝐼) ∧ 𝑠 ∈ (𝒫 𝑆 ∩ Fin)) → 𝑠 ∈ 𝐼) |
| 11 | 10 | ralrimiva 3146 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ 𝐼) → ∀𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑠 ∈ 𝐼) |
| 12 | | dfss3 3972 |
. . 3
⊢
((𝒫 𝑆 ∩
Fin) ⊆ 𝐼 ↔
∀𝑠 ∈ (𝒫
𝑆 ∩ Fin)𝑠 ∈ 𝐼) |
| 13 | 11, 12 | sylibr 234 |
. 2
⊢ ((𝜑 ∧ 𝑆 ∈ 𝐼) → (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) |
| 14 | 2 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → 𝐴 ∈ (Moore‘𝑋)) |
| 15 | | acsfiindd.4 |
. . . 4
⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
| 16 | 15 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → 𝑆 ⊆ 𝑋) |
| 17 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) |
| 18 | | elfpw 9394 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin) ↔ (𝑡 ⊆ (𝑆 ∖ {𝑥}) ∧ 𝑡 ∈ Fin)) |
| 19 | 17, 18 | sylib 218 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → (𝑡 ⊆ (𝑆 ∖ {𝑥}) ∧ 𝑡 ∈ Fin)) |
| 20 | 19 | simpld 494 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → 𝑡 ⊆ (𝑆 ∖ {𝑥})) |
| 21 | 20 | difss2d 4139 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → 𝑡 ⊆ 𝑆) |
| 22 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → 𝑥 ∈ 𝑆) |
| 23 | 22 | snssd 4809 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → {𝑥} ⊆ 𝑆) |
| 24 | 21, 23 | unssd 4192 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → (𝑡 ∪ {𝑥}) ⊆ 𝑆) |
| 25 | 19 | simprd 495 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → 𝑡 ∈ Fin) |
| 26 | | snfi 9083 |
. . . . . . . . . . . 12
⊢ {𝑥} ∈ Fin |
| 27 | | unfi 9211 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ Fin ∧ {𝑥} ∈ Fin) → (𝑡 ∪ {𝑥}) ∈ Fin) |
| 28 | 25, 26, 27 | sylancl 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → (𝑡 ∪ {𝑥}) ∈ Fin) |
| 29 | | elfpw 9394 |
. . . . . . . . . . 11
⊢ ((𝑡 ∪ {𝑥}) ∈ (𝒫 𝑆 ∩ Fin) ↔ ((𝑡 ∪ {𝑥}) ⊆ 𝑆 ∧ (𝑡 ∪ {𝑥}) ∈ Fin)) |
| 30 | 24, 28, 29 | sylanbrc 583 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → (𝑡 ∪ {𝑥}) ∈ (𝒫 𝑆 ∩ Fin)) |
| 31 | 2 | ad4antr 732 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → 𝐴 ∈ (Moore‘𝑋)) |
| 32 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → 𝑠 ∈ 𝐼) |
| 33 | | simpllr 776 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑥 ∈ 𝑆) |
| 34 | | snidg 4660 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ {𝑥}) |
| 35 | | elun2 4183 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ (𝑡 ∪ {𝑥})) |
| 36 | 33, 34, 35 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑥 ∈ (𝑡 ∪ {𝑥})) |
| 37 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑠 = (𝑡 ∪ {𝑥})) |
| 38 | 36, 37 | eleqtrrd 2844 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑥 ∈ 𝑠) |
| 39 | 38 | adantr 480 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → 𝑥 ∈ 𝑠) |
| 40 | 4, 5, 31, 32, 39 | ismri2dad 17680 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))) |
| 41 | 2 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝐴 ∈ (Moore‘𝑋)) |
| 42 | 20 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑡 ⊆ (𝑆 ∖ {𝑥})) |
| 43 | | neldifsnd 4793 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → ¬ 𝑥 ∈ (𝑆 ∖ {𝑥})) |
| 44 | 42, 43 | ssneldd 3986 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → ¬ 𝑥 ∈ 𝑡) |
| 45 | | difsnb 4806 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 ∈ 𝑡 ↔ (𝑡 ∖ {𝑥}) = 𝑡) |
| 46 | 44, 45 | sylib 218 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑡 ∖ {𝑥}) = 𝑡) |
| 47 | | ssun1 4178 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑡 ⊆ (𝑡 ∪ {𝑥}) |
| 48 | 47, 37 | sseqtrrid 4027 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑡 ⊆ 𝑠) |
| 49 | 48 | ssdifd 4145 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑡 ∖ {𝑥}) ⊆ (𝑠 ∖ {𝑥})) |
| 50 | 46, 49 | eqsstrrd 4019 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑡 ⊆ (𝑠 ∖ {𝑥})) |
| 51 | 24 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑡 ∪ {𝑥}) ⊆ 𝑆) |
| 52 | 15 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑆 ⊆ 𝑋) |
| 53 | 51, 52 | sstrd 3994 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑡 ∪ {𝑥}) ⊆ 𝑋) |
| 54 | 37, 53 | eqsstrd 4018 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → 𝑠 ⊆ 𝑋) |
| 55 | 54 | ssdifssd 4147 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑠 ∖ {𝑥}) ⊆ 𝑋) |
| 56 | 41, 4, 50, 55 | mrcssd 17667 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑁‘𝑡) ⊆ (𝑁‘(𝑠 ∖ {𝑥}))) |
| 57 | 56 | sseld 3982 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑥 ∈ (𝑁‘𝑡) → 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) |
| 58 | 57 | adantr 480 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → (𝑥 ∈ (𝑁‘𝑡) → 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})))) |
| 59 | 40, 58 | mtod 198 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) ∧ 𝑠 ∈ 𝐼) → ¬ 𝑥 ∈ (𝑁‘𝑡)) |
| 60 | 59 | ex 412 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) ∧ 𝑠 = (𝑡 ∪ {𝑥})) → (𝑠 ∈ 𝐼 → ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 61 | 30, 60 | rspcimdv 3612 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → (∀𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑠 ∈ 𝐼 → ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 62 | 12, 61 | biimtrid 242 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)) → ((𝒫 𝑆 ∩ Fin) ⊆ 𝐼 → ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 63 | 62 | impancom 451 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → (𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin) → ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 64 | 63 | ralrimiv 3145 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → ∀𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin) ¬ 𝑥 ∈ (𝑁‘𝑡)) |
| 65 | 15 | ssdifssd 4147 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
| 66 | 1, 4, 65 | acsficl2d 18597 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∃𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)𝑥 ∈ (𝑁‘𝑡))) |
| 67 | 66 | notbid 318 |
. . . . . . . 8
⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ¬ ∃𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)𝑥 ∈ (𝑁‘𝑡))) |
| 68 | | ralnex 3072 |
. . . . . . . 8
⊢
(∀𝑡 ∈
(𝒫 (𝑆 ∖
{𝑥}) ∩ Fin) ¬ 𝑥 ∈ (𝑁‘𝑡) ↔ ¬ ∃𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin)𝑥 ∈ (𝑁‘𝑡)) |
| 69 | 67, 68 | bitr4di 289 |
. . . . . . 7
⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∀𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin) ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 70 | 69 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ ∀𝑡 ∈ (𝒫 (𝑆 ∖ {𝑥}) ∩ Fin) ¬ 𝑥 ∈ (𝑁‘𝑡))) |
| 71 | 64, 70 | mpbird 257 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
| 72 | 71 | an32s 652 |
. . . 4
⊢ (((𝜑 ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑆) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
| 73 | 72 | ralrimiva 3146 |
. . 3
⊢ ((𝜑 ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
| 74 | 4, 5, 14, 16, 73 | ismri2dd 17677 |
. 2
⊢ ((𝜑 ∧ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼) → 𝑆 ∈ 𝐼) |
| 75 | 13, 74 | impbida 801 |
1
⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼)) |