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Mirrors > Home > MPE Home > Th. List > ssopab2b | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker ssopab2bw 5436 when possible. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ssopab2b | ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopab1 5137 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab1 5137 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 1, 2 | nfss 3962 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
4 | nfopab2 5138 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | nfopab2 5138 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜓} | |
6 | 4, 5 | nfss 3962 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} |
7 | ssel 3963 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | |
8 | opabid 5415 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | opabid 5415 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜓) | |
10 | 7, 8, 9 | 3imtr3g 297 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → (𝜑 → 𝜓)) |
11 | 6, 10 | alrimi 2213 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑦(𝜑 → 𝜓)) |
12 | 3, 11 | alrimi 2213 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} → ∀𝑥∀𝑦(𝜑 → 𝜓)) |
13 | ssopab2 5435 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
14 | 12, 13 | impbii 211 | 1 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ⊆ wss 3938 〈cop 4575 {copab 5130 |
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-13 2390 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 |
This theorem is referenced by: eqopab2b 5441 |
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