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| Mirrors > Home > MPE Home > Th. List > difprsnss | Structured version Visualization version 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 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | elpr 4650 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
| 3 | velsn 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 4 | 3 | notbii 320 | . . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴) |
| 5 | biorf 937 | . . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))) | |
| 6 | 5 | biimparc 479 | . . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
| 7 | 2, 4, 6 | syl2anb 598 | . . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵) |
| 8 | eldif 3961 | . . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴})) | |
| 9 | velsn 4642 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
| 10 | 7, 8, 9 | 3imtr4i 292 | . 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵}) |
| 11 | 10 | ssriv 3987 | 1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: en2other2 10049 pmtrprfv 19471 itg11 25726 |
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