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Theorem rrx2pnecoorneor 44071
Description: If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then they are different at least at one coordinate. (Contributed by AV, 26-Feb-2023.)
Hypotheses
Ref Expression
rrx2pnecoorneor.i 𝐼 = {1, 2}
rrx2pnecoorneor.b 𝑃 = (ℝ ↑𝑚 𝐼)
Assertion
Ref Expression
rrx2pnecoorneor ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))

Proof of Theorem rrx2pnecoorneor
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . . . . 7 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
2 rrx2pnecoorneor.i . . . . . . . . 9 𝐼 = {1, 2}
32raleqi 3353 . . . . . . . 8 (∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖) ↔ ∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖))
4 1ex 10437 . . . . . . . . 9 1 ∈ V
5 2ex 11520 . . . . . . . . 9 2 ∈ V
6 fveq2 6501 . . . . . . . . . 10 (𝑖 = 1 → (𝑋𝑖) = (𝑋‘1))
7 fveq2 6501 . . . . . . . . . 10 (𝑖 = 1 → (𝑌𝑖) = (𝑌‘1))
86, 7eqeq12d 2793 . . . . . . . . 9 (𝑖 = 1 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘1) = (𝑌‘1)))
9 fveq2 6501 . . . . . . . . . 10 (𝑖 = 2 → (𝑋𝑖) = (𝑋‘2))
10 fveq2 6501 . . . . . . . . . 10 (𝑖 = 2 → (𝑌𝑖) = (𝑌‘2))
119, 10eqeq12d 2793 . . . . . . . . 9 (𝑖 = 2 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘2) = (𝑌‘2)))
124, 5, 8, 11ralpr 4511 . . . . . . . 8 (∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
133, 12bitri 267 . . . . . . 7 (∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
141, 13sylibr 226 . . . . . 6 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖))
15 elmapfn 8231 . . . . . . . . . 10 (𝑋 ∈ (ℝ ↑𝑚 𝐼) → 𝑋 Fn 𝐼)
16 rrx2pnecoorneor.b . . . . . . . . . 10 𝑃 = (ℝ ↑𝑚 𝐼)
1715, 16eleq2s 2884 . . . . . . . . 9 (𝑋𝑃𝑋 Fn 𝐼)
18 elmapfn 8231 . . . . . . . . . 10 (𝑌 ∈ (ℝ ↑𝑚 𝐼) → 𝑌 Fn 𝐼)
1918, 16eleq2s 2884 . . . . . . . . 9 (𝑌𝑃𝑌 Fn 𝐼)
2017, 19anim12i 603 . . . . . . . 8 ((𝑋𝑃𝑌𝑃) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
2120adantr 473 . . . . . . 7 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
22 eqfnfv 6629 . . . . . . 7 ((𝑋 Fn 𝐼𝑌 Fn 𝐼) → (𝑋 = 𝑌 ↔ ∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖)))
2321, 22syl 17 . . . . . 6 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 = 𝑌 ↔ ∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖)))
2414, 23mpbird 249 . . . . 5 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → 𝑋 = 𝑌)
2524ex 405 . . . 4 ((𝑋𝑃𝑌𝑃) → (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)) → 𝑋 = 𝑌))
2625necon3ad 2980 . . 3 ((𝑋𝑃𝑌𝑃) → (𝑋𝑌 → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))))
27263impia 1097 . 2 ((𝑋𝑃𝑌𝑃𝑋𝑌) → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
28 neorian 3062 . 2 (((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)) ↔ ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
2927, 28sylibr 226 1 ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wral 3088  {cpr 4444   Fn wfn 6185  cfv 6190  (class class class)co 6978  𝑚 cmap 8208  cr 10336  1c1 10338  2c2 11498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-1cn 10395  ax-addcl 10397
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-map 8210  df-2 11506
This theorem is referenced by:  rrx2pnedifcoorneor  44072  inlinecirc02p  44143
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