Theorem relssi 3244

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 groups:   x,y,A   x,B,y

Proof of Theorem relssi

Step	Hyp	Ref	Expression
1		relssi.1	. . 3 ⊢ Rel A
2		ssrel 3243	. . 3 ⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
3	1, 2	ax-mp 7	. 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 962	. 2 ⊢ ∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)
6	3, 5	mpgbir 987	1 ⊢ A ⊆ B

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 953   ∈ wcel 957   ⊆ wss 2044  ⟨cop 2408  Rel wrel 3171

This theorem is referenced by:  xpsspw 3253  resiexg 3392  oprssdm 4037  ecopoprdm 4302  enssdom 4373  idssen 4396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-opab 2663  df-xp 3180  df-rel 3181

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