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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 3558 | . 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵) | |
2 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | elint2 4417 | . . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
4 | 3 | ralbii 2963 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
5 | ralcom 3079 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
6 | dfss3 3558 | . . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
7 | 6 | ralbii 2963 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) |
8 | 5, 7 | bitr4i 266 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
9 | 1, 4, 8 | 3bitri 285 | 1 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ∩ cint 4410 |
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-v 3175 df-in 3547 df-ss 3554 df-int 4411 |
This theorem is referenced by: ssintab 4429 ssintub 4430 iinpw 4550 trint 4696 oneqmini 5693 fint 5997 sorpssint 6845 iscard2 8685 coftr 8978 isf32lem2 9059 inttsk 9475 dfrtrcl2 13650 isacs1i 16141 mrelatglb 17007 fbfinnfr 21455 fclscmp 21644 fneint 31513 topmeet 31529 igenval2 33035 ismrcd1 36279 dftrcl3 37031 dfrtrcl3 37044 sssalgen 39229 issalgend 39232 |
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