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Theorem unixpid 5632
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 5093 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2 0xp 5165 . . . 4 (∅ × 𝐴) = ∅
31, 2syl6eq 2676 . . 3 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4 unieq 4415 . . . . 5 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
54unieqd 4417 . . . 4 ((𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = ∅)
6 uni0 4436 . . . . . 6 ∅ = ∅
76unieqi 4416 . . . . 5 ∅ =
87, 6eqtri 2648 . . . 4 ∅ = ∅
9 eqtr 2645 . . . . 5 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 × 𝐴) = ∅)
10 eqtr 2645 . . . . . . 7 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → (𝐴 × 𝐴) = 𝐴)
1110expcom 451 . . . . . 6 (∅ = 𝐴 → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
1211eqcoms 2634 . . . . 5 (𝐴 = ∅ → ( (𝐴 × 𝐴) = ∅ → (𝐴 × 𝐴) = 𝐴))
139, 12syl5com 31 . . . 4 (( (𝐴 × 𝐴) = ∅ ∧ ∅ = ∅) → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
145, 8, 13sylancl 693 . . 3 ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴))
153, 14mpcom 38 . 2 (𝐴 = ∅ → (𝐴 × 𝐴) = 𝐴)
16 df-ne 2797 . . 3 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17 xpnz 5516 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18 unixp 5630 . . . . 5 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = (𝐴𝐴))
19 unidm 3739 . . . . 5 (𝐴𝐴) = 𝐴
2018, 19syl6eq 2676 . . . 4 ((𝐴 × 𝐴) ≠ ∅ → (𝐴 × 𝐴) = 𝐴)
2117, 20sylbi 207 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → (𝐴 × 𝐴) = 𝐴)
2216, 16, 21sylancbr 699 . 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 384   = wceq 1480  wne 2796  cun 3558  c0 3896   cuni 4407   × cxp 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090
This theorem is referenced by:  psss  17130
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