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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 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | elpr 3612 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
3 | velsn 3608 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | 3 | notbii 668 | . . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴) |
5 | biorf 744 | . . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))) | |
6 | 5 | biimparc 299 | . . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
7 | 2, 4, 6 | syl2anb 291 | . . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵) |
8 | eldif 3138 | . . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴})) | |
9 | velsn 3608 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
10 | 7, 8, 9 | 3imtr4i 201 | . 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵}) |
11 | 10 | ssriv 3159 | 1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∖ cdif 3126 ⊆ wss 3129 {csn 3591 {cpr 3592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 |
This theorem is referenced by: en2other2 7189 |
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