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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 5542 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 27-Dec-1996.) Avoid ax-13 2370. (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 5211 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 5211 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 1, 2 | nfss 3970 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
4 | nfopab2 5212 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | nfopab2 5212 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜓} | |
6 | 4, 5 | nfss 3970 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
7 | ssel 3971 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
8 | opabidw 5517 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | opabidw 5517 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜓) | |
10 | 7, 8, 9 | 3imtr3g 294 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
11 | 6, 10 | alrimi 2206 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑦(𝜑 → 𝜓)) |
12 | 3, 11 | alrimi 2206 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑥∀𝑦(𝜑 → 𝜓)) |
13 | ssopab2 5539 | . 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 1539 ∈ wcel 2106 ⊆ wss 3944 〈cop 4628 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-opab 5204 |
This theorem is referenced by: eqopab2bw 5541 dffun2OLDOLD 6544 marypha2lem3 9414 cvmlift2lem12 34136 cossssid2 37143 |
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