Theorem ssopab2i 2818

Description: Inference of ordered pair abstraction subclass from implication.

Hypothesis

Ref	Expression
ssopab2i.1	⊢ (φ → ψ)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem ssopab2i

Step	Hyp	Ref	Expression
1		ssopab2 2817	. 2 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} ↔ ∀x∀y(φ → ψ))
2		ssopab2i.1	. . 3 ⊢ (φ → ψ)
3	2	ax-gen 961	. 2 ⊢ ∀y(φ → ψ)
4	1, 3	mpgbir 986	1 ⊢ {⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ}

Syntax hints:   → wi 3  ∀wal 952   ⊆ wss 2043  {copab 2661

This theorem is referenced by:  opabssxp 3229  relopab 3261  tz7.44-1 3919  tz7.44-2 3920  tz7.44-3 3921  ssoprab2i 3999  eloprabi 4108  aceq3 4713  dfef2 7257  infmap2lem2 7530  bcthlem15 7963  nvvcop 8165  ajfval 8413  cmpfun 10399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662

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