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Mirrors > Home > MPE Home > Th. List > ss2rabi | Structured version Visualization version GIF version |
Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
ss2rabi.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
ss2rabi | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rab 3641 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) | |
2 | ss2rabi.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprgbir 2911 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 |
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-ral 2901 df-rab 2905 df-in 3547 df-ss 3554 |
This theorem is referenced by: supub 8248 suplub 8249 card2on 8342 rankval4 8613 fin1a2lem12 9116 catlid 16167 catrid 16168 gsumval2 17103 lbsextlem3 18981 psrbagsn 19316 musum 24717 ppiub 24729 umgrupgr 25769 umgrislfupgr 25789 usisuslgra 25894 disjxwwlks 26264 clwlknclwlkdifnum 26488 omssubadd 29689 bj-unrab 32114 poimirlem26 32605 poimirlem27 32606 lclkrs2 35847 usgruspgr 40408 usgrislfuspgr 40414 disjxwwlksn 41110 clwwlknclwwlkdifnum 41182 konigsbergssiedgw 41419 |
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