Theorem propssopi 4971

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.)

Hypotheses

Ref	Expression
snopeqop.a	⊢ 𝐴 ∈ V
snopeqop.b	⊢ 𝐵 ∈ V
snopeqop.c	⊢ 𝐶 ∈ V
snopeqop.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 4401	. . 3 ⊢ ⟨𝐸, 𝐹⟩ = {{𝐸}, {𝐸, 𝐹}}
4	3	sseq2i 3630	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}})
5		sspr 4366	. . 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 4932	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	prnz 4310	. . . . . 6 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ≠ ∅
8		eqneqall 2805	. . . . . 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 4932	. . . . . . 7 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
11		snex 4908	. . . . . . 7 ⊢ {𝐸} ∈ V
12	6, 10, 11	preqsn 4393	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸}))
13		snopeqop.a	. . . . . . . . 9 ⊢ 𝐴 ∈ V
14		snopeqop.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
15	13, 14	opth 4945	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
16		simpl 473	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 𝐴 = 𝐶)
17	15, 16	sylbi 207	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
18	17	adantr 481	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸}) → 𝐴 = 𝐶)
19	12, 18	sylbi 207	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}} → 𝐴 = 𝐶)
20	9, 19	jaoi 394	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ∅ ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}}) → 𝐴 = 𝐶)
21		prex 4909	. . . . . . 7 ⊢ {𝐸, 𝐹} ∈ V
22	6, 10, 21	preqsn 4393	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}))
23	16	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} → 𝐴 = 𝐶))
24	15, 23	sylbi 207	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} → 𝐴 = 𝐶))
25	24	imp 445	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) → 𝐴 = 𝐶)
26	22, 25	sylbi 207	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} → 𝐴 = 𝐶)
27	3	eqcomi 2631	. . . . . . . 8 ⊢ {{𝐸}, {𝐸, 𝐹}} = ⟨𝐸, 𝐹⟩
28	27	eqeq2i 2634	. . . . . . 7 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩)
29		snopeqop.c	. . . . . . . 8 ⊢ 𝐶 ∈ V
30		snopeqop.d	. . . . . . . 8 ⊢ 𝐷 ∈ V
31	13, 14, 29, 30, 1, 2	propeqop 4970	. . . . . . 7 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
32	28, 31	bitri 264	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
33		simpll 790	. . . . . 6 ⊢ (((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))) → 𝐴 = 𝐶)
34	32, 33	sylbi 207	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}} → 𝐴 = 𝐶)
35	26, 34	jaoi 394	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸, 𝐹}} ∨ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {{𝐸}, {𝐸, 𝐹}}) → 𝐴 = 𝐶)
36	20, 35	jaoi 394	. . 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)
37	5, 36	sylbi 207	. 2 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ {{𝐸}, {𝐸, 𝐹}} → 𝐴 = 𝐶)
38	4, 37	sylbi 207	1 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  iunopeqop  4981