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Mirrors > Home > ILE Home > Th. List > ssoprab2b | GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4013. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2b | ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfoprab1 5554 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | nfoprab1 5554 | . . . 4 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
3 | 1, 2 | nfss 2938 | . . 3 ⊢ Ⅎ𝑥{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
4 | nfoprab2 5555 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | nfoprab2 5555 | . . . . 5 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
6 | 4, 5 | nfss 2938 | . . . 4 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
7 | nfoprab3 5556 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
8 | nfoprab3 5556 | . . . . . 6 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} | |
9 | 7, 8 | nfss 2938 | . . . . 5 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
10 | ssel 2939 | . . . . . 6 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → 〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓})) | |
11 | oprabid 5537 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜑) | |
12 | oprabid 5537 | . . . . . 6 ⊢ (〈〈𝑥, 𝑦〉, 𝑧〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ 𝜓) | |
13 | 10, 11, 12 | 3imtr3g 193 | . . . . 5 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
14 | 9, 13 | alrimi 1415 | . . . 4 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑧(𝜑 → 𝜓)) |
15 | 6, 14 | alrimi 1415 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑦∀𝑧(𝜑 → 𝜓)) |
16 | 3, 15 | alrimi 1415 | . 2 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} → ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
17 | ssoprab2 5561 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) | |
18 | 16, 17 | impbii 117 | 1 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 ∈ wcel 1393 ⊆ wss 2917 〈cop 3378 {coprab 5513 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-oprab 5516 |
This theorem is referenced by: eqoprab2b 5563 |
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