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Mirrors > Home > ILE Home > Th. List > xpdom3m | GIF version |
Description: A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.) |
Ref | Expression |
---|---|
xpdom3m | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∃𝑥 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsneng 6788 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≈ 𝐴) | |
2 | 1 | 3adant2 1006 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≈ 𝐴) |
3 | 2 | ensymd 6749 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≈ (𝐴 × {𝑥})) |
4 | xpexg 4718 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | |
5 | 4 | 3adant3 1007 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × 𝐵) ∈ V) |
6 | simp3 989 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | 6 | snssd 3718 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵) |
8 | xpss2 4715 | . . . . . . 7 ⊢ ({𝑥} ⊆ 𝐵 → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵)) | |
9 | 7, 8 | syl 14 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵)) |
10 | ssdomg 6744 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × {𝑥}) ⊆ (𝐴 × 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵))) | |
11 | 5, 9, 10 | sylc 62 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) |
12 | endomtr 6756 | . . . . 5 ⊢ ((𝐴 ≈ (𝐴 × {𝑥}) ∧ (𝐴 × {𝑥}) ≼ (𝐴 × 𝐵)) → 𝐴 ≼ (𝐴 × 𝐵)) | |
13 | 3, 11, 12 | syl2anc 409 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵)) |
14 | 13 | 3expia 1195 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 × 𝐵))) |
15 | 14 | exlimdv 1807 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 × 𝐵))) |
16 | 15 | 3impia 1190 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∃𝑥 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 968 ∃wex 1480 ∈ wcel 2136 Vcvv 2726 ⊆ wss 3116 {csn 3576 class class class wbr 3982 × cxp 4602 ≈ cen 6704 ≼ cdom 6705 |
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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-er 6501 df-en 6707 df-dom 6708 |
This theorem is referenced by: (None) |
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