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Mirrors > Home > MPE Home > Th. List > ssopab2bw | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5550 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 27-Dec-1996.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
ssopab2bw | ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopab1 5219 | . . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | nfopab1 5219 | . . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜓} | |
3 | 1, 2 | nfss 3975 | . . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} |
4 | nfopab2 5220 | . . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
5 | nfopab2 5220 | . . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜓} | |
6 | 4, 5 | nfss 3975 | . . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} |
7 | ssel 3976 | . . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})) | |
8 | opabidw 5525 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) | |
9 | opabidw 5525 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜓) | |
10 | 7, 8, 9 | 3imtr3g 295 | . . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (𝜑 → 𝜓)) |
11 | 6, 10 | alrimi 2207 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑦(𝜑 → 𝜓)) |
12 | 3, 11 | alrimi 2207 | . 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑥∀𝑦(𝜑 → 𝜓)) |
13 | ssopab2 5547 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) | |
14 | 12, 13 | impbii 208 | 1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∈ wcel 2107 ⊆ wss 3949 ⟨cop 4635 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 |
This theorem is referenced by: eqopab2bw 5549 dffun2OLDOLD 6556 marypha2lem3 9432 cvmlift2lem12 34305 cossssid2 37338 |
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