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Mirrors > Home > ILE Home > Th. List > ssint | 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 2989 | . 2 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵) | |
2 | vex 2604 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | elint2 3643 | . . 3 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
4 | 3 | ralbii 2372 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
5 | ralcom 2517 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
6 | dfss3 2989 | . . . 4 ⊢ (𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) | |
7 | 6 | ralbii 2372 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝑥) |
8 | 5, 7 | bitr4i 185 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
9 | 1, 4, 8 | 3bitri 204 | 1 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∈ wcel 1433 ∀wral 2348 ⊆ wss 2973 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-in 2979 df-ss 2986 df-int 3637 |
This theorem is referenced by: ssintab 3653 ssintub 3654 iinpw 3763 trint 3890 fintm 5095 bj-ssom 10731 |
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