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Mirrors > Home > ILE Home > Th. List > 2oneel | GIF version |
Description: ∅ and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.) |
Ref | Expression |
---|---|
2oneel | ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 6461 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 2445 | . 2 ⊢ ∅ ≠ 1o |
3 | 0lt2o 6470 | . . 3 ⊢ ∅ ∈ 2o | |
4 | 1lt2o 6471 | . . 3 ⊢ 1o ∈ 2o | |
5 | neeq1 2373 | . . . 4 ⊢ (𝑢 = ∅ → (𝑢 ≠ 𝑣 ↔ ∅ ≠ 𝑣)) | |
6 | neeq2 2374 | . . . 4 ⊢ (𝑣 = 1o → (∅ ≠ 𝑣 ↔ ∅ ≠ 1o)) | |
7 | 5, 6 | opelopab2 4291 | . . 3 ⊢ ((∅ ∈ 2o ∧ 1o ∈ 2o) → (⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ↔ ∅ ≠ 1o)) |
8 | 3, 4, 7 | mp2an 426 | . 2 ⊢ (⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ↔ ∅ ≠ 1o) |
9 | 2, 8 | mpbir 146 | 1 ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 ⟨cop 3613 {copab 4081 1oc1o 6438 2oc2o 6439 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-opab 4083 df-tr 4120 df-iord 4387 df-on 4389 df-suc 4392 df-1o 6445 df-2o 6446 |
This theorem is referenced by: 2omotaplemst 7292 |
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