Theorem relssi 2481

Description: Inference from subclass principle for relations.

Hypotheses

Ref	Expression
relssi.1	⊢ Rel A
relssi.2	⊢ (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)

Assertion

Ref	Expression
relssi	⊢ A ⊆ B

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem relssi

Step	Hyp	Ref	Expression
1		relssi.1	. . 3 ⊢ Rel A
2		relss 2480	. . 3 ⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
3	1, 2	ax-mp 6	. 2 ⊢ (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B))
4		relssi.2	. . 3 ⊢ (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)
5	4	ax-gen 677	. 2 ⊢ ∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)
6	3, 5	mpgbir 686	1 ⊢ A ⊆ B

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810  Rel wrel 2415

This theorem is referenced by:  xpex 2488  oprssdm 3056  ecopoprdm 3245  enssdom 3287  idssen 3309

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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