Theorem eqrelrel 5763

Description: Extensionality principle for ordered triples (used by 2-place operations df-oprab 7394), analogous to eqrel 5750. Use relrelss 6249 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Assertion

Ref	Expression
eqrelrel	⊢ ((𝐴 ∪ 𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem eqrelrel

Step	Hyp	Ref	Expression
1		unss 4156	. 2 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ 𝐵 ⊆ ((V × V) × V)) ↔ (𝐴 ∪ 𝐵) ⊆ ((V × V) × V))
2		ssrelrel 5762	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
3		ssrelrel 5762	. . . 4 ⊢ (𝐵 ⊆ ((V × V) × V) → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)))
4	2, 3	bi2anan9 638	. . 3 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ 𝐵 ⊆ ((V × V) × V)) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))))
5		eqss 3965	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
6		2albiim 1890	. . . . 5 ⊢ (∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ↔ (∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)))
7	6	albii 1819	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ↔ ∀𝑥(∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)))
8		19.26 1870	. . . 4 ⊢ (∀𝑥(∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)) ↔ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)))
9	7, 8	bitri 275	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ↔ (∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴)))
10	4, 5, 9	3bitr4g 314	. 2 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ 𝐵 ⊆ ((V × V) × V)) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
11	1, 10	sylbir 235	1 ⊢ ((𝐴 ∪ 𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ⟨cop 4598   × cxp 5639

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647

This theorem is referenced by: (None)