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Mirrors > Home > MPE Home > Th. List > ssopab2 | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
Ref | Expression |
---|---|
ssopab2 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | anim2d 587 | . . . . 5 ⊢ ((𝜑 → 𝜓) → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
3 | 2 | aleximi 1749 | . . . 4 ⊢ (∀𝑦(𝜑 → 𝜓) → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
4 | 3 | aleximi 1749 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
5 | 4 | ss2abdv 3638 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
6 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
7 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
8 | 5, 6, 7 | 3sstr4g 3609 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 {cab 2596 ⊆ wss 3540 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-opab 4644 |
This theorem is referenced by: ssopab2b 4927 ssopab2i 4928 ssopab2dv 4929 opabbrex 6593 |
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