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Mirrors > Home > ILE Home > Th. List > disjne | GIF version |
Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
disjne | ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 3486 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
2 | eleq1 2252 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵)) | |
3 | 2 | notbid 668 | . . . . 5 ⊢ (𝑥 = 𝐶 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵)) |
4 | 3 | rspccva 2855 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵) |
5 | eleq1a 2261 | . . . . 5 ⊢ (𝐷 ∈ 𝐵 → (𝐶 = 𝐷 → 𝐶 ∈ 𝐵)) | |
6 | 5 | necon3bd 2403 | . . . 4 ⊢ (𝐷 ∈ 𝐵 → (¬ 𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐷)) |
7 | 4, 6 | syl5com 29 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷)) |
8 | 1, 7 | sylanb 284 | . 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝐶 ≠ 𝐷)) |
9 | 8 | 3impia 1202 | 1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 ∩ cin 3143 ∅c0 3437 |
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-3an 982 df-tru 1367 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-v 2754 df-dif 3146 df-in 3150 df-nul 3438 |
This theorem is referenced by: (None) |
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