xpopth - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer	< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xpopth		Structured version GIF version

Theorem xpopth 5744

Description: An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

**Assertion**
Ref	Expression
xpopth	⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (((1^st ‘A) = (1^st ‘B) ∧ (2^nd ‘A) = (2^nd ‘B)) ↔ A = B))

**Proof of Theorem xpopth**
Step	Hyp	Ref	Expression
1		1st2nd2 5743	. . 3 ⊢ (A ∈ (𝐶 × 𝐷) → A = ⟨(1^st ‘A), (2^nd ‘A)⟩)
2		1st2nd2 5743	. . 3 ⊢ (B ∈ (𝑅 × 𝑆) → B = ⟨(1^st ‘B), (2^nd ‘B)⟩)
3	1, 2	eqeqan12d 2052	. 2 ⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (A = B ↔ ⟨(1^st ‘A), (2^nd ‘A)⟩ = ⟨(1^st ‘B), (2^nd ‘B)⟩))
4		1stexg 5736	. . . 4 ⊢ (A ∈ (𝐶 × 𝐷) → (1^st ‘A) ∈ V)
5		2ndexg 5737	. . . 4 ⊢ (A ∈ (𝐶 × 𝐷) → (2^nd ‘A) ∈ V)
6		opthg 3966	. . . 4 ⊢ (((1^st ‘A) ∈ V ∧ (2^nd ‘A) ∈ V) → (⟨(1^st ‘A), (2^nd ‘A)⟩ = ⟨(1^st ‘B), (2^nd ‘B)⟩ ↔ ((1^st ‘A) = (1^st ‘B) ∧ (2^nd ‘A) = (2^nd ‘B))))
7	4, 5, 6	syl2anc 391	. . 3 ⊢ (A ∈ (𝐶 × 𝐷) → (⟨(1^st ‘A), (2^nd ‘A)⟩ = ⟨(1^st ‘B), (2^nd ‘B)⟩ ↔ ((1^st ‘A) = (1^st ‘B) ∧ (2^nd ‘A) = (2^nd ‘B))))
8	7	adantr 261	. 2 ⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (⟨(1^st ‘A), (2^nd ‘A)⟩ = ⟨(1^st ‘B), (2^nd ‘B)⟩ ↔ ((1^st ‘A) = (1^st ‘B) ∧ (2^nd ‘A) = (2^nd ‘B))))
9	3, 8	bitr2d 178	1 ⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (((1^st ‘A) = (1^st ‘B) ∧ (2^nd ‘A) = (2^nd ‘B)) ↔ A = B))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⟨cop 3370 × cxp 4286 ‘cfv 4845 1^st c1st 5707 2^nd c2nd 5708

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fo 4851 df-fv 4853 df-1st 5709 df-2nd 5710

This theorem is referenced by: (None)