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Mirrors > Home > ILE Home > Th. List > xpss12 | GIF version |
Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
xpss12 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3141 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | ssel 3141 | . . . 4 ⊢ (𝐶 ⊆ 𝐷 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) | |
3 | 1, 2 | im2anan9 593 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) |
4 | 3 | ssopab2dv 4263 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)}) |
5 | df-xp 4617 | . 2 ⊢ (𝐴 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} | |
6 | df-xp 4617 | . 2 ⊢ (𝐵 × 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)} | |
7 | 4, 5, 6 | 3sstr4g 3190 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ⊆ wss 3121 {copab 4049 × cxp 4609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 df-opab 4051 df-xp 4617 |
This theorem is referenced by: xpss 4719 xpss1 4721 xpss2 4722 djussxp 4756 ssxpbm 5046 ssrnres 5053 cossxp 5133 cossxp2 5134 cocnvss 5136 relrelss 5137 fssxp 5365 oprabss 5939 pmss12g 6653 caserel 7064 casef 7065 dmaddpi 7287 dmmulpi 7288 rexpssxrxp 7964 ltrelxr 7980 dfz2 9284 phimullem 12179 txuni2 13050 txbas 13052 neitx 13062 txcnp 13065 cnmpt2res 13091 psmetres2 13127 xmetres2 13173 metres2 13175 xmetresbl 13234 xmettx 13304 qtopbasss 13315 tgqioo 13341 resubmet 13342 limccnp2lem 13439 limccnp2cntop 13440 |
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