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Mirrors > Home > MPE Home > Th. List > ssoprab2 | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5433. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2 | ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . 7 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | anim2d 613 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
3 | 2 | aleximi 1832 | . . . . 5 ⊢ (∀𝑧(𝜑 → 𝜓) → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
4 | 3 | aleximi 1832 | . . . 4 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
5 | 4 | aleximi 1832 | . . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
6 | 5 | ss2abdv 4044 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} ⊆ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)}) |
7 | df-oprab 7160 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
8 | df-oprab 7160 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
9 | 6, 7, 8 | 3sstr4g 4012 | 1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 {cab 2799 ⊆ wss 3936 〈cop 4573 {coprab 7157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-oprab 7160 |
This theorem is referenced by: ssoprab2b 7223 eqoprab2bw 7224 joinfval 17611 meetfval 17625 |
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