Theorem rrx2pnecoorneor 49749

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	⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))

Proof of Theorem rrx2pnecoorneor

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrx2pnecoorneor.i	. . . . . . . . 9 ⊢ 𝐼 = {1, 2}
2	1	raleqi 3317	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑖 ∈ {1, 2} (𝑋‘𝑖) = (𝑌‘𝑖))
3		1ex 11275	. . . . . . . . 9 ⊢ 1 ∈ V
4		2ex 12390	. . . . . . . . 9 ⊢ 2 ∈ V
5		fveq2 6873	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑋‘𝑖) = (𝑋‘1))
6		fveq2 6873	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑌‘𝑖) = (𝑌‘1))
7	5, 6	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘1) = (𝑌‘1)))
8		fveq2 6873	. . . . . . . . . 10 ⊢ (𝑖 = 2 → (𝑋‘𝑖) = (𝑋‘2))
9		fveq2 6873	. . . . . . . . . 10 ⊢ (𝑖 = 2 → (𝑌‘𝑖) = (𝑌‘2))
10	8, 9	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝑖 = 2 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘2) = (𝑌‘2)))
11	3, 4, 7, 10	ralpr 4660	. . . . . . . 8 ⊢ (∀𝑖 ∈ {1, 2} (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
12	2, 11	bitri 278	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
13	12	bilanri 512	. . . . . 6 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖))
14		elmapfn 8865	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → 𝑋 Fn 𝐼)
15		rrx2pnecoorneor.b	. . . . . . . . . 10 ⊢ 𝑃 = (ℝ ↑m 𝐼)
16	14, 15	eleq2s 2878	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → 𝑋 Fn 𝐼)
17		elmapfn 8865	. . . . . . . . . 10 ⊢ (𝑌 ∈ (ℝ ↑m 𝐼) → 𝑌 Fn 𝐼)
18	17, 15	eleq2s 2878	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑃 → 𝑌 Fn 𝐼)
19	16, 18	anim12i 625	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼))
20	19	adantr 486	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼))
21		eqfnfv 7017	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖)))
22	20, 21	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝐼 (𝑋‘𝑖) = (𝑌‘𝑖)))
23	13, 22	mpbird 260	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))) → 𝑋 = 𝑌)
24	23	ex 418	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)) → 𝑋 = 𝑌))
25	24	necon3ad 2968	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≠ 𝑌 → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2))))
26	25	3impia 1135	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
27		neorian 3050	. 2 ⊢ (((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)) ↔ ¬ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) = (𝑌‘2)))
28	26, 27	sylibr 237	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  {cpr 4585   Fn wfn 6522  ‘cfv 6527  (class class class)co 7408   ↑_m cmap 8825  ℝcr 11171  1c1 11173  2c2 12367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-1cn 11230  ax-addcl 11232

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-map 8827  df-2 12375

This theorem is used by:  rrx2pnedifcoorneor  49750  inlinecirc02p  49821