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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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | elpr 4146 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
3 | velsn 4141 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | 3 | notbii 309 | . . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴) |
5 | biorf 419 | . . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))) | |
6 | 5 | biimparc 503 | . . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
7 | 2, 4, 6 | syl2anb 495 | . . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵) |
8 | eldif 3550 | . . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴})) | |
9 | velsn 4141 | . . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
10 | 7, 8, 9 | 3imtr4i 280 | . 2 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) → 𝑥 ∈ {𝐵}) |
11 | 10 | ssriv 3572 | 1 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 |
This theorem is referenced by: en2other2 8715 pmtrprfv 17696 itg11 23264 |
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