Theorem propssopi 5363

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 4762	. . 3 ⊢ ⟨𝐸, 𝐹⟩ = {{𝐸}, {𝐸, 𝐹}}
4	3	sseq2i 3944	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}})
5		sspr 4726	. . 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 5321	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	prnz 4673	. . . . . 6 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ≠ ∅
8		eqneqall 2998	. . . . . 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 5321	. . . . . . 7 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
11	6, 10	preqsn 4752	. . . . . 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 5333	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
15		simpl 486	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐶)
16	14, 15	sylbi 220	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
17	16	adantr 484	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸}) → 𝐴 = 𝐶)
18	11, 17	sylbi 220	. . . . 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 4752	. . . . . 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 220	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} → 𝐴 = 𝐶))
23	22	imp 410	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) → 𝐴 = 𝐶)
24	20, 23	sylbi 220	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} → 𝐴 = 𝐶)
25	3	eqcomi 2807	. . . . . . . 8 ⊢ {{𝐸}, {𝐸, 𝐹}} = ⟨𝐸, 𝐹⟩
26	25	eqeq2i 2811	. . . . . . 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 5362	. . . . . . 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 278	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
31		simpll 766	. . . . . 6 ⊢ (((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))) → 𝐴 = 𝐶)
32	30, 31	sylbi 220	. . . . 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 220	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}} → 𝐴 = 𝐶)
36	4, 35	sylbi 220	1 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  ⟨cop 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532

This theorem is referenced by:  iunopeqop  5376