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Mirrors > Home > ILE Home > Th. List > difprsn1 | GIF version |
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
difprsn1 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necom 2444 | . 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵) | |
2 | df-pr 3617 | . . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | equncomi 3296 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
4 | 3 | difeq1i 3264 | . . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 3517 | . . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2210 | . . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) |
7 | disjsn2 3673 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅) | |
8 | disj3 3490 | . . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴})) | |
9 | 7, 8 | sylib 122 | . . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴})) |
10 | 6, 9 | eqtr4id 2241 | . 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
11 | 1, 10 | sylbir 135 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ≠ wne 2360 ∖ cdif 3141 ∪ cun 3142 ∩ cin 3143 ∅c0 3437 {csn 3610 {cpr 3611 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3616 df-pr 3617 |
This theorem is referenced by: difprsn2 3750 |
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