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Mirrors > Home > ILE Home > Th. List > difprsnss | GIF version |
Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
difprsnss | ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | elpr 3543 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
3 | velsn 3539 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | 3 | notbii 657 | . . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴) |
5 | biorf 733 | . . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))) | |
6 | 5 | biimparc 297 | . . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
7 | 2, 4, 6 | syl2anb 289 | . . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵) |
8 | eldif 3075 | . . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴})) | |
9 | velsn 3539 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
10 | 7, 8, 9 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵}) |
11 | 10 | ssriv 3096 | 1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∖ cdif 3063 ⊆ wss 3066 {csn 3522 {cpr 3523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 |
This theorem is referenced by: en2other2 7045 |
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