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Theorem unixpid 6314
Description: Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.)
Assertion
Ref Expression
unixpid (𝐴 × 𝐴) = 𝐴

Proof of Theorem unixpid
StepHypRef Expression
1 xpeq1 5713 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2 0xp 5797 . . . 4 (∅ × 𝐴) = ∅
31, 2eqtrdi 2790 . . 3 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4 unieq 4942 . . . . 5 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
54unieqd 4944 . . . 4 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
6 uni0 4961 . . . . . 6 ∅ = ∅
76unieqi 4943 . . . . 5 ∅ =
87, 6eqtri 2762 . . . 4 ∅ = ∅
9 eqtr 2757 . . . . 5 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 × 𝐴) = ∅)
10 eqtr 2757 . . . . . . 7 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → (𝐴 × 𝐴) = 𝐴)
1110expcom 413 . . . . . 6 (∅ = 𝐴 → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
1211eqcoms 2742 . . . . 5 (𝐴 = ∅ → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
139, 12syl5com 31 . . . 4 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
145, 8, 13sylancl 585 . . 3 ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
153, 14mpcom 38 . 2 (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴)
16 df-ne 2943 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17 xpnz 6189 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18 unixp 6312 . . . . 5 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = (𝐴𝐴))
19 unidm 4174 . . . . 5 (𝐴𝐴) = 𝐴
2018, 19eqtrdi 2790 . . . 4 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = 𝐴)
2117, 20sylbi 217 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → (𝐴 × 𝐴) = 𝐴)
2216, 16, 21sylancbr 600 . 2 𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴)
2315, 22pm2.61i 182 1 (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wne 2942  cun 3968  c0 4347   cuni 4931   × cxp 5697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710
This theorem is referenced by:  psss  18645
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