Theorem ssopab2i 2120

Description: Inference of abstraction subclass from implication.

Hypothesis

Ref	Expression
ssopab2i.1	⊢ (φ → ψ)

Assertion

Ref	Expression
ssopab2i	⊢ {⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ}

Distinct variable group(s):   x,y

Proof of Theorem ssopab2i

Step	Hyp	Ref	Expression
1		ssopab2 2119	. 2 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} ↔ ∀x∀y(φ → ψ))
2		ssopab2i.1	. . 3 ⊢ (φ → ψ)
3	2	ax-gen 677	. 2 ⊢ ∀y(φ → ψ)
4	1, 3	mpgbir 686	1 ⊢ {⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ}

Syntax hints:   → wi 2  ∀wal 672   ⊆ wss 1487  {copab 2055

This theorem is referenced by:  opabssxp 2468  relopab 2494  tz7.44-1 2966  tz7.44-2 2967  tz7.44-3 2968  ssoprab2i 3036  aceq3 3556  infmap2lem2 4952

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

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