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Mirrors > Home > MPE Home > Th. List > rabss2 | Structured version Visualization version GIF version |
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rabss2 | ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.45 623 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
2 | 1 | alimi 1814 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
3 | dfss2 3969 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | ss2ab 4057 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
5 | 2, 3, 4 | 3imtr4i 292 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
7 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
8 | 5, 6, 7 | 3sstr4g 4028 | 1 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 {cab 2710 {crab 3433 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: rabssrabd 4082 sess2 5646 hashbcss 16937 dprdss 19899 minveclem4 24949 prmdvdsfi 26611 mumul 26685 sqff1o 26686 rpvmasumlem 26990 disjxwwlkn 29167 clwwlknfi 29298 shatomistici 31614 rabfodom 31743 xpinpreima2 32887 ballotth 33536 bj-unrab 35806 icorempo 36232 lssats 37882 lpssat 37883 lssatle 37885 lssat 37886 atlatmstc 38189 dochspss 40249 rmxyelqirrOLD 41649 idomodle 41938 sssmf 45454 |
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