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Mirrors > Home > MPE Home > Th. List > ssint | Structured version Visualization version GIF version |
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
ssint | ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3971 | . 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵) | |
2 | vex 3479 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | elint2 4958 | . . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
4 | 3 | ralbii 3094 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
5 | ralcom 3287 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
6 | dfss3 3971 | . . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
7 | 6 | ralbii 3094 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) |
8 | 5, 7 | bitr4i 278 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
9 | 1, 4, 8 | 3bitri 297 | 1 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3949 ∩ cint 4951 |
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-11 2155 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3956 df-ss 3966 df-int 4952 |
This theorem is referenced by: ssintab 4970 ssintub 4971 iinpw 5110 oneqmini 6417 fint 6771 fnssintima 7359 sorpssint 7723 iscard2 9971 coftr 10268 isf32lem2 10349 inttsk 10769 dfrtrcl2 15009 isacs1i 17601 mrelatglb 18513 fbfinnfr 23345 fclscmp 23534 noextenddif 27171 eqscut2 27307 scutun12 27311 ssmxidllem 32589 fneint 35233 topmeet 35249 igenval2 36934 ismrcd1 41436 onintunirab 41976 dftrcl3 42471 dfrtrcl3 42484 sssalgen 45051 issalgend 45054 intubeu 47609 ipoglblem 47614 |
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