Theorem funsndifnop 7171

Description: A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
funsndifnop.a	⊢ 𝐴 ∈ V
funsndifnop.b	⊢ 𝐵 ∈ V
funsndifnop.g	⊢ 𝐺 = {⟨𝐴, 𝐵⟩}

Assertion

Ref	Expression
funsndifnop	⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))

Proof of Theorem funsndifnop

Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5760	. . 3 ⊢ (𝐺 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩)
2		funsndifnop.g	. . . . . 6 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
3		funsndifnop.a	. . . . . . . 8 ⊢ 𝐴 ∈ V
4		funsndifnop.b	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	3, 4	funsn 6619	. . . . . . 7 ⊢ Fun {⟨𝐴, 𝐵⟩}
6		funeq 6586	. . . . . . 7 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩}))
7	5, 6	mpbiri 258	. . . . . 6 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺)
8	2, 7	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
9		funeq 6586	. . . . . . 7 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10		vex 3484	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 3484	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	funop 7169	. . . . . . 7 ⊢ (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))
13	9, 12	bitrdi 287	. . . . . 6 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})))
14		eqeq2 2749	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16		opex 5469	. . . . . . . . . . . . . . 15 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
17	16	sneqr 4840	. . . . . . . . . . . . . 14 ⊢ ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
18	3, 4	opth 5481	. . . . . . . . . . . . . . 15 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎 ∧ 𝐵 = 𝑎))
19		eqtr3 2763	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → 𝐴 = 𝐵)
20	19	a1d 25	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵))
21	18, 20	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))
22	17, 21	syl 17	. . . . . . . . . . . . 13 ⊢ ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
23	15, 22	biimtrdi 253	. . . . . . . . . . . 12 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
24	2, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺 = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
25	14, 24	biimtrdi 253	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
26	25	com23 86	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)))
27	26	impcom 407	. . . . . . . 8 ⊢ ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
28	27	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
29	28	com12 32	. . . . . 6 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → 𝐴 = 𝐵))
30	13, 29	sylbid 240	. . . . 5 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 → 𝐴 = 𝐵))
31	8, 30	mpi 20	. . . 4 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
32	31	exlimivv 1932	. . 3 ⊢ (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
33	1, 32	sylbi 217	. 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
34	33	necon3ai 2965	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  {csn 4626  ⟨cop 4632   × cxp 5683  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569

This theorem is referenced by:  funsneqopb  7172  snstrvtxval  29054  snstriedgval  29055