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Theorem unixpid 5831
Description: Field of a square Cartesian product. (Contributed by FL, 10-Oct-2009.)
Assertion
Ref Expression
unixpid (𝐴 × 𝐴) = 𝐴

Proof of Theorem unixpid
StepHypRef Expression
1 xpeq1 5280 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2 0xp 5356 . . . 4 (∅ × 𝐴) = ∅
31, 2syl6eq 2810 . . 3 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4 unieq 4596 . . . . 5 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
54unieqd 4598 . . . 4 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
6 uni0 4617 . . . . . 6 ∅ = ∅
76unieqi 4597 . . . . 5 ∅ =
87, 6eqtri 2782 . . . 4 ∅ = ∅
9 eqtr 2779 . . . . 5 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 × 𝐴) = ∅)
10 eqtr 2779 . . . . . . 7 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → (𝐴 × 𝐴) = 𝐴)
1110expcom 450 . . . . . 6 (∅ = 𝐴 → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
1211eqcoms 2768 . . . . 5 (𝐴 = ∅ → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
139, 12syl5com 31 . . . 4 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
145, 8, 13sylancl 697 . . 3 ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
153, 14mpcom 38 . 2 (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴)
16 df-ne 2933 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17 xpnz 5711 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18 unixp 5829 . . . . 5 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = (𝐴𝐴))
19 unidm 3899 . . . . 5 (𝐴𝐴) = 𝐴
2018, 19syl6eq 2810 . . . 4 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = 𝐴)
2117, 20sylbi 207 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → (𝐴 × 𝐴) = 𝐴)
2216, 16, 21sylancbr 703 . 2 𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴)
2315, 22pm2.61i 176 1 (𝐴 × 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wne 2932  cun 3713  c0 4058   cuni 4588   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  psss  17415
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