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Theorem rrx2pnecoorneor 48103
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 483 . . . . . . 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 3313 . . . . . . . 8 (∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖) ↔ ∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖))
4 1ex 11260 . . . . . . . . 9 1 ∈ V
5 2ex 12341 . . . . . . . . 9 2 ∈ V
6 fveq2 6901 . . . . . . . . . 10 (𝑖 = 1 → (𝑋𝑖) = (𝑋‘1))
7 fveq2 6901 . . . . . . . . . 10 (𝑖 = 1 → (𝑌𝑖) = (𝑌‘1))
86, 7eqeq12d 2742 . . . . . . . . 9 (𝑖 = 1 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘1) = (𝑌‘1)))
9 fveq2 6901 . . . . . . . . . 10 (𝑖 = 2 → (𝑋𝑖) = (𝑋‘2))
10 fveq2 6901 . . . . . . . . . 10 (𝑖 = 2 → (𝑌𝑖) = (𝑌‘2))
119, 10eqeq12d 2742 . . . . . . . . 9 (𝑖 = 2 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘2) = (𝑌‘2)))
124, 5, 8, 11ralpr 4709 . . . . . . . 8 (∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
133, 12bitri 274 . . . . . . 7 (∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
141, 13sylibr 233 . . . . . 6 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖))
15 elmapfn 8894 . . . . . . . . . 10 (𝑋 ∈ (ℝ ↑m 𝐼) → 𝑋 Fn 𝐼)
16 rrx2pnecoorneor.b . . . . . . . . . 10 𝑃 = (ℝ ↑m 𝐼)
1715, 16eleq2s 2844 . . . . . . . . 9 (𝑋𝑃𝑋 Fn 𝐼)
18 elmapfn 8894 . . . . . . . . . 10 (𝑌 ∈ (ℝ ↑m 𝐼) → 𝑌 Fn 𝐼)
1918, 16eleq2s 2844 . . . . . . . . 9 (𝑌𝑃𝑌 Fn 𝐼)
2017, 19anim12i 611 . . . . . . . 8 ((𝑋𝑃𝑌𝑃) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
2120adantr 479 . . . . . . 7 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
22 eqfnfv 7044 . . . . . . 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 256 . . . . 5 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → 𝑋 = 𝑌)
2524ex 411 . . . 4 ((𝑋𝑃𝑌𝑃) → (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)) → 𝑋 = 𝑌))
2625necon3ad 2943 . . 3 ((𝑋𝑃𝑌𝑃) → (𝑋𝑌 → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))))
27263impia 1114 . 2 ((𝑋𝑃𝑌𝑃𝑋𝑌) → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
28 neorian 3027 . 2 (((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)) ↔ ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
2927, 28sylibr 233 1 ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  {cpr 4635   Fn wfn 6549  cfv 6554  (class class class)co 7424  m cmap 8855  cr 11157  1c1 11159  2c2 12319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-1cn 11216  ax-addcl 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-2 12327
This theorem is referenced by:  rrx2pnedifcoorneor  48104  inlinecirc02p  48175
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