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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 3479 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | elpr 4652 | . . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
3 | velsn 4645 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | 3 | notbii 320 | . . . 4 ⊢ (¬ 𝑥 ∈ {𝐴} ↔ ¬ 𝑥 = 𝐴) |
5 | biorf 936 | . . . . 5 ⊢ (¬ 𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))) | |
6 | 5 | biimparc 481 | . . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∧ ¬ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
7 | 2, 4, 6 | syl2anb 599 | . . 3 ⊢ ((𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴}) → 𝑥 = 𝐵) |
8 | eldif 3959 | . . 3 ⊢ (𝑥 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ↔ (𝑥 ∈ {𝐴, 𝐵} ∧ ¬ 𝑥 ∈ {𝐴})) | |
9 | velsn 4645 | . . 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ⊆ wss 3949 {csn 4629 {cpr 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-sn 4630 df-pr 4632 |
This theorem is referenced by: en2other2 10004 pmtrprfv 19321 itg11 25208 |
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