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Theorem rrx2pnecoorneor 47401
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 486 . . . . . . 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 3324 . . . . . . . 8 (∀𝑖𝐼 (𝑋𝑖) = (𝑌𝑖) ↔ ∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖))
4 1ex 11210 . . . . . . . . 9 1 ∈ V
5 2ex 12289 . . . . . . . . 9 2 ∈ V
6 fveq2 6892 . . . . . . . . . 10 (𝑖 = 1 → (𝑋𝑖) = (𝑋‘1))
7 fveq2 6892 . . . . . . . . . 10 (𝑖 = 1 → (𝑌𝑖) = (𝑌‘1))
86, 7eqeq12d 2749 . . . . . . . . 9 (𝑖 = 1 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘1) = (𝑌‘1)))
9 fveq2 6892 . . . . . . . . . 10 (𝑖 = 2 → (𝑋𝑖) = (𝑋‘2))
10 fveq2 6892 . . . . . . . . . 10 (𝑖 = 2 → (𝑌𝑖) = (𝑌‘2))
119, 10eqeq12d 2749 . . . . . . . . 9 (𝑖 = 2 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋‘2) = (𝑌‘2)))
124, 5, 8, 11ralpr 4705 . . . . . . . 8 (∀𝑖 ∈ {1, 2} (𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
133, 12bitri 275 . . . . . . 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 8859 . . . . . . . . . 10 (𝑋 ∈ (ℝ ↑m 𝐼) → 𝑋 Fn 𝐼)
16 rrx2pnecoorneor.b . . . . . . . . . 10 𝑃 = (ℝ ↑m 𝐼)
1715, 16eleq2s 2852 . . . . . . . . 9 (𝑋𝑃𝑋 Fn 𝐼)
18 elmapfn 8859 . . . . . . . . . 10 (𝑌 ∈ (ℝ ↑m 𝐼) → 𝑌 Fn 𝐼)
1918, 16eleq2s 2852 . . . . . . . . 9 (𝑌𝑃𝑌 Fn 𝐼)
2017, 19anim12i 614 . . . . . . . 8 ((𝑋𝑃𝑌𝑃) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
2120adantr 482 . . . . . . 7 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 Fn 𝐼𝑌 Fn 𝐼))
22 eqfnfv 7033 . . . . . . 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 257 . . . . 5 (((𝑋𝑃𝑌𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → 𝑋 = 𝑌)
2524ex 414 . . . 4 ((𝑋𝑃𝑌𝑃) → (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)) → 𝑋 = 𝑌))
2625necon3ad 2954 . . 3 ((𝑋𝑃𝑌𝑃) → (𝑋𝑌 → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))))
27263impia 1118 . 2 ((𝑋𝑃𝑌𝑃𝑋𝑌) → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
28 neorian 3038 . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  {cpr 4631   Fn wfn 6539  cfv 6544  (class class class)co 7409  m cmap 8820  cr 11109  1c1 11111  2c2 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-2 12275
This theorem is referenced by:  rrx2pnedifcoorneor  47402  inlinecirc02p  47473
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