Theorem funsndifnop 7148

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 5736	. . 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 6589	. . . . . . 7 ⊢ Fun {⟨𝐴, 𝐵⟩}
6		funeq 6556	. . . . . . 7 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (Fun 𝐺 ↔ Fun {⟨𝐴, 𝐵⟩}))
7	5, 6	mpbiri 261	. . . . . 6 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → Fun 𝐺)
8	2, 7	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
9		funeq 6556	. . . . . . 7 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ Fun ⟨𝑥, 𝑦⟩))
10		vex 3457	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	funop 7146	. . . . . . 7 ⊢ (Fun ⟨𝑥, 𝑦⟩ ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}))
13	9, 12	bitrdi 290	. . . . . 6 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 ↔ ∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩})))
14		eqeq2 2773	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ ↔ 𝐺 = {⟨𝑎, 𝑎⟩}))
15		eqeq1 2765	. . . . . . . . . . . . 13 ⊢ (𝐺 = {⟨𝐴, 𝐵⟩} → (𝐺 = {⟨𝑎, 𝑎⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩}))
16		opex 5445	. . . . . . . . . . . . . . 15 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
17	16	sneqr 4804	. . . . . . . . . . . . . 14 ⊢ ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩)
18	3, 4	opth 5458	. . . . . . . . . . . . . . 15 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ ↔ (𝐴 = 𝑎 ∧ 𝐵 = 𝑎))
19		eqtr3 2783	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → 𝐴 = 𝐵)
20	19	a1d 26	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑎) → (𝑥 = {𝑎} → 𝐴 = 𝐵))
21	18, 20	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑎⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵))
22	17, 21	syl 18	. . . . . . . . . . . . 13 ⊢ ({⟨𝐴, 𝐵⟩} = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → 𝐴 = 𝐵))
23	15, 22	biimtrdi 256	. . . . . . . . . . . 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 256	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝐺 = ⟨𝑥, 𝑦⟩ → (𝑥 = {𝑎} → 𝐴 = 𝐵)))
26	25	com23 87	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩} → (𝑥 = {𝑎} → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)))
27	26	impcom 412	. . . . . . . 8 ⊢ ((𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
28	27	exlimiv 1958	. . . . . . 7 ⊢ (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵))
29	28	com12 33	. . . . . 6 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (∃𝑎(𝑥 = {𝑎} ∧ ⟨𝑥, 𝑦⟩ = {⟨𝑎, 𝑎⟩}) → 𝐴 = 𝐵))
30	13, 29	sylbid 243	. . . . 5 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → (Fun 𝐺 → 𝐴 = 𝐵))
31	8, 30	mpi 21	. . . 4 ⊢ (𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
32	31	exlimivv 1960	. . 3 ⊢ (∃𝑥∃𝑦 𝐺 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝐵)
33	1, 32	sylbi 220	. 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵)
34	33	necon3ai 2981	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  {csn 4588  ⟨cop 4594   × cxp 5659  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  funsneqopb  7149  snstrvtxval  29353  snstriedgval  29354