Theorem propssopi 5508

Description: If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 16-Jun-2022.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
snopeqop.a	⊢ 𝐴 ∈ V
snopeqop.b	⊢ 𝐵 ∈ V
propeqop.c	⊢ 𝐶 ∈ V
propeqop.d	⊢ 𝐷 ∈ V
propeqop.e	⊢ 𝐸 ∈ V
propeqop.f	⊢ 𝐹 ∈ V

Assertion

Ref	Expression
propssopi	⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Proof of Theorem propssopi

Step	Hyp	Ref	Expression
1		propeqop.e	. . . 4 ⊢ 𝐸 ∈ V
2		propeqop.f	. . . 4 ⊢ 𝐹 ∈ V
3	1, 2	dfop 4872	. . 3 ⊢ ⟨𝐸, 𝐹⟩ = {{𝐸}, {𝐸, 𝐹}}
4	3	sseq2i 4011	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}})
5		sspr 4836	. . 3 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}} ↔ (({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}}) ∨ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}})))
6		opex 5464	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	prnz 4781	. . . . . 6 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ≠ ∅
8		eqneqall 2950	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ≠ ∅ → 𝐴 = 𝐶))
9	7, 8	mpi 20	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ → 𝐴 = 𝐶)
10		opex 5464	. . . . . . 7 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
11	6, 10	preqsn 4862	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸}))
12		snopeqop.a	. . . . . . . . 9 ⊢ 𝐴 ∈ V
13		snopeqop.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
14	12, 13	opth 5476	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
15		simpl 482	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐶)
16	14, 15	sylbi 216	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
17	16	adantr 480	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸}) → 𝐴 = 𝐶)
18	11, 17	sylbi 216	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}} → 𝐴 = 𝐶)
19	9, 18	jaoi 854	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}}) → 𝐴 = 𝐶)
20	6, 10	preqsn 4862	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}))
21	15	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} → 𝐴 = 𝐶))
22	14, 21	sylbi 216	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} → 𝐴 = 𝐶))
23	22	imp 406	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) → 𝐴 = 𝐶)
24	20, 23	sylbi 216	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} → 𝐴 = 𝐶)
25	3	eqcomi 2740	. . . . . . . 8 ⊢ {{𝐸}, {𝐸, 𝐹}} = ⟨𝐸, 𝐹⟩
26	25	eqeq2i 2744	. . . . . . 7 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩)
27		propeqop.c	. . . . . . . 8 ⊢ 𝐶 ∈ V
28		propeqop.d	. . . . . . . 8 ⊢ 𝐷 ∈ V
29	12, 13, 27, 28, 1, 2	propeqop 5507	. . . . . . 7 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
30	26, 29	bitri 275	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
31		simpll 764	. . . . . 6 ⊢ (((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))) → 𝐴 = 𝐶)
32	30, 31	sylbi 216	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} → 𝐴 = 𝐶)
33	24, 32	jaoi 854	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}}) → 𝐴 = 𝐶)
34	19, 33	jaoi 854	. . 3 ⊢ ((({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}}) ∨ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}})) → 𝐴 = 𝐶)
35	5, 34	sylbi 216	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}} → 𝐴 = 𝐶)
36	4, 35	sylbi 216	1 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473   ⊆ wss 3948  ∅c0 4322  {csn 4628  {cpr 4630  ⟨cop 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635

This theorem is referenced by:  iunopeqop  5521