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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 4047 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) | |
2 | ss2rabi.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprgbir 3153 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {crab 3142 ⊆ wss 3936 |
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-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-in 3943 df-ss 3952 |
This theorem is referenced by: f1ossf1o 6890 supub 8923 suplub 8924 card2on 9018 rankval4 9296 fin1a2lem12 9833 catlid 16954 catrid 16955 gsumval2 17896 lbsextlem3 19932 psrbagsn 20275 musum 25768 ppiub 25780 umgrupgr 26888 umgrislfupgr 26908 usgruspgr 26963 usgrislfuspgr 26969 disjxwwlksn 27682 clwwlknclwwlkdifnum 27758 konigsbergssiedgw 28029 omssubadd 31558 bj-unrab 34247 poimirlem26 34933 poimirlem27 34934 ssrabi 35526 lclkrs2 38691 |
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