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Mirrors > Home > MPE Home > Th. List > ssdifsn | Structured version Visualization version GIF version |
Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) |
Ref | Expression |
---|---|
ssdifsn | ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3592 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝐵 ∖ {𝐶})) | |
2 | eldifsn 4317 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶)) | |
3 | 2 | ralbii 2980 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶)) |
4 | 1, 3 | bitri 264 | . . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶)) |
5 | r19.26 3064 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶)) | |
6 | 4, 5 | bitri 264 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶)) |
7 | dfss3 3592 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | |
8 | 7 | bicomi 214 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ 𝐴 ⊆ 𝐵) |
9 | neirr 2803 | . . . . 5 ⊢ ¬ 𝐶 ≠ 𝐶 | |
10 | neeq1 2856 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ≠ 𝐶 ↔ 𝐶 ≠ 𝐶)) | |
11 | 10 | rspccv 3306 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 → (𝐶 ∈ 𝐴 → 𝐶 ≠ 𝐶)) |
12 | 9, 11 | mtoi 190 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 → ¬ 𝐶 ∈ 𝐴) |
13 | nelelne 2892 | . . . . 5 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ≠ 𝐶)) | |
14 | 13 | ralrimiv 2965 | . . . 4 ⊢ (¬ 𝐶 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶) |
15 | 12, 14 | impbii 199 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶 ↔ ¬ 𝐶 ∈ 𝐴) |
16 | 8, 15 | anbi12i 733 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
17 | 6, 16 | bitri 264 | 1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-sn 4178 |
This theorem is referenced by: logdivsqrle 30728 elsetrecslem 42444 |
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