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Theorem ssopab2b 4041
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
ssopab2b ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Proof of Theorem ssopab2b
StepHypRef Expression
1 nfopab1 3854 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 3854 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜓}
31, 2nfss 2966 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
4 nfopab2 3855 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 3855 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜓}
64, 5nfss 2966 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
7 ssel 2967 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
8 opabid 4022 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
9 opabid 4022 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜓)
107, 8, 93imtr3g 197 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (𝜑𝜓))
116, 10alrimi 1431 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑦(𝜑𝜓))
123, 11alrimi 1431 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑥𝑦(𝜑𝜓))
13 ssopab2 4040 . 2 (∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
1412, 13impbii 121 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wcel 1409  wss 2945  cop 3406  {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847
This theorem is referenced by:  eqopab2b  4044  dffun2  4940
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