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Mirrors > Home > ILE Home > Th. List > xpid11m | GIF version |
Description: The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.) |
Ref | Expression |
---|---|
xpid11m | ⊢ ((∃x x ∈ A ∧ ∃x x ∈ B) → ((A × A) = (B × B) ↔ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmxpm 4498 | . . . . . 6 ⊢ (∃x x ∈ A → dom (A × A) = A) | |
2 | 1 | adantr 261 | . . . . 5 ⊢ ((∃x x ∈ A ∧ ∃x x ∈ B) → dom (A × A) = A) |
3 | dmeq 4478 | . . . . 5 ⊢ ((A × A) = (B × B) → dom (A × A) = dom (B × B)) | |
4 | 2, 3 | sylan9req 2090 | . . . 4 ⊢ (((∃x x ∈ A ∧ ∃x x ∈ B) ∧ (A × A) = (B × B)) → A = dom (B × B)) |
5 | dmxpm 4498 | . . . . 5 ⊢ (∃x x ∈ B → dom (B × B) = B) | |
6 | 5 | ad2antlr 458 | . . . 4 ⊢ (((∃x x ∈ A ∧ ∃x x ∈ B) ∧ (A × A) = (B × B)) → dom (B × B) = B) |
7 | 4, 6 | eqtrd 2069 | . . 3 ⊢ (((∃x x ∈ A ∧ ∃x x ∈ B) ∧ (A × A) = (B × B)) → A = B) |
8 | 7 | ex 108 | . 2 ⊢ ((∃x x ∈ A ∧ ∃x x ∈ B) → ((A × A) = (B × B) → A = B)) |
9 | xpeq12 4307 | . . 3 ⊢ ((A = B ∧ A = B) → (A × A) = (B × B)) | |
10 | 9 | anidms 377 | . 2 ⊢ (A = B → (A × A) = (B × B)) |
11 | 8, 10 | impbid1 130 | 1 ⊢ ((∃x x ∈ A ∧ ∃x x ∈ B) → ((A × A) = (B × B) ↔ A = B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 × cxp 4286 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-dm 4298 |
This theorem is referenced by: (None) |
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