![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > xpdom1g | GIF version |
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
xpdom1g | ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6748 | . . . 4 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 4671 | . . 3 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
3 | xpcomeng 6831 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑉) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴)) | |
4 | 3 | ancoms 268 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴)) |
5 | 2, 4 | sylan2 286 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≈ (𝐶 × 𝐴)) |
6 | xpdom2g 6835 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵)) | |
7 | 1 | brrelex2i 4672 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V) |
8 | xpcomeng 6831 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ∈ V) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) | |
9 | 7, 8 | sylan2 286 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) |
10 | domentr 6794 | . . 3 ⊢ (((𝐶 × 𝐴) ≼ (𝐶 × 𝐵) ∧ (𝐶 × 𝐵) ≈ (𝐵 × 𝐶)) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) | |
11 | 6, 9, 10 | syl2anc 411 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) |
12 | endomtr 6793 | . 2 ⊢ (((𝐴 × 𝐶) ≈ (𝐶 × 𝐴) ∧ (𝐶 × 𝐴) ≼ (𝐵 × 𝐶)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
13 | 5, 11, 12 | syl2anc 411 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 Vcvv 2739 class class class wbr 4005 × cxp 4626 ≈ cen 6741 ≼ cdom 6742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-1st 6144 df-2nd 6145 df-en 6744 df-dom 6745 |
This theorem is referenced by: xpdom1 6838 xpct 12400 |
Copyright terms: Public domain | W3C validator |