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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 5548. (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 610 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
3 | 2 | aleximi 1826 | . . . . 5 ⊢ (∀𝑧(𝜑 → 𝜓) → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
4 | 3 | aleximi 1826 | . . . 4 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓))) |
5 | 4 | aleximi 1826 | . . 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 4057 | . 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 7423 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
8 | df-oprab 7423 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
9 | 6, 7, 8 | 3sstr4g 4022 | 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 394 ∀wal 1531 = wceq 1533 ∃wex 1773 {cab 2702 ⊆ wss 3944 〈cop 4636 {coprab 7420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-ss 3961 df-oprab 7423 |
This theorem is referenced by: ssoprab2b 7489 eqoprab2bw 7490 joinfval 18368 meetfval 18382 |
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