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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 5507 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Dec-1996.) Avoid ax-13 2371. (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 5176 | . . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
2 | nfopab1 5176 | . . . 4 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜓} | |
3 | 1, 2 | nfss 3937 | . . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} |
4 | nfopab2 5177 | . . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
5 | nfopab2 5177 | . . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜓} | |
6 | 4, 5 | nfss 3937 | . . . 4 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} |
7 | ssel 3938 | . . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})) | |
8 | opabidw 5482 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) | |
9 | opabidw 5482 | . . . . 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 5504 | . 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 3911 ⟨cop 4593 {copab 5168 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 |
This theorem is referenced by: eqopab2bw 5506 dffun2OLDOLD 6509 marypha2lem3 9374 cvmlift2lem12 33911 cossssid2 36933 |
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