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| Mirrors > Home > MPE Home > Th. List > Mathboxes > untsucf | Structured version Visualization version GIF version | ||
| Description: If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| Ref | Expression |
|---|---|
| untsucf.1 | ⊢ Ⅎ𝑦𝐴 |
| Ref | Expression |
|---|---|
| untsucf | ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | untsucf.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦 ¬ 𝑥 ∈ 𝑥 | |
| 3 | 1, 2 | nfralw 3311 | . 2 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 |
| 4 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 5 | 4 | elsuc 6454 | . . 3 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) |
| 6 | elequ1 2115 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
| 7 | elequ2 2123 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) | |
| 8 | 6, 7 | bitrd 279 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
| 9 | 8 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
| 10 | 9 | rspccv 3619 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ 𝐴 → ¬ 𝑦 ∈ 𝑦)) |
| 11 | untelirr 35708 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴) | |
| 12 | eleq1 2829 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)) | |
| 13 | eleq2 2830 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴)) | |
| 14 | 12, 13 | bitrd 279 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴)) |
| 15 | 14 | notbid 318 | . . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐴)) |
| 16 | 11, 15 | syl5ibrcom 247 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 = 𝐴 → ¬ 𝑦 ∈ 𝑦)) |
| 17 | 10, 16 | jaod 860 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → ¬ 𝑦 ∈ 𝑦)) |
| 18 | 5, 17 | biimtrid 242 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦 ∈ 𝑦)) |
| 19 | 3, 18 | ralrimi 3257 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 suc csuc 6386 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-v 3482 df-un 3956 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: dfon2lem3 35786 |
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