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Theorem unixpid 6274
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 5681 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2 0xp 5765 . . . 4 (∅ × 𝐴) = ∅
31, 2eqtrdi 2780 . . 3 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4 unieq 4911 . . . . 5 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
54unieqd 4913 . . . 4 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
6 uni0 4930 . . . . . 6 ∅ = ∅
76unieqi 4912 . . . . 5 ∅ =
87, 6eqtri 2752 . . . 4 ∅ = ∅
9 eqtr 2747 . . . . 5 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 × 𝐴) = ∅)
10 eqtr 2747 . . . . . . 7 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → (𝐴 × 𝐴) = 𝐴)
1110expcom 413 . . . . . 6 (∅ = 𝐴 → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
1211eqcoms 2732 . . . . 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 2933 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17 xpnz 6149 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18 unixp 6272 . . . . 5 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = (𝐴𝐴))
19 unidm 4145 . . . . 5 (𝐴𝐴) = 𝐴
2018, 19eqtrdi 2780 . . . 4 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = 𝐴)
2117, 20sylbi 216 . . 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 1533  wne 2932  cun 3939  c0 4315   cuni 4900   × cxp 5665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678
This theorem is referenced by:  psss  18537
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