Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > unxpdom | Structured version Visualization version GIF version |
Description: Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
unxpdom | ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 8504 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5602 | . . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
3 | 1 | brrelex2i 5602 | . . 3 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
4 | 2, 3 | anim12i 612 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | breq2 5061 | . . . . 5 ⊢ (𝑥 = 𝐴 → (1o ≺ 𝑥 ↔ 1o ≺ 𝐴)) | |
6 | 5 | anbi1d 629 | . . . 4 ⊢ (𝑥 = 𝐴 → ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝑦))) |
7 | uneq1 4129 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
8 | xpeq1 5562 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦)) | |
9 | 7, 8 | breq12d 5070 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))) |
10 | 6, 9 | imbi12d 346 | . . 3 ⊢ (𝑥 = 𝐴 → (((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))) |
11 | breq2 5061 | . . . . 5 ⊢ (𝑦 = 𝐵 → (1o ≺ 𝑦 ↔ 1o ≺ 𝐵)) | |
12 | 11 | anbi2d 628 | . . . 4 ⊢ (𝑦 = 𝐵 → ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝐵))) |
13 | uneq2 4130 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
14 | xpeq2 5569 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵)) | |
15 | 13, 14 | breq12d 5070 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
16 | 12, 15 | imbi12d 346 | . . 3 ⊢ (𝑦 = 𝐵 → (((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))) |
17 | eqid 2818 | . . . 4 ⊢ (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉)) = (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉)) | |
18 | eqid 2818 | . . . 4 ⊢ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉) = if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉) | |
19 | 17, 18 | unxpdomlem3 8712 | . . 3 ⊢ ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) |
20 | 10, 16, 19 | vtocl2g 3569 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
21 | 4, 20 | mpcom 38 | 1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ifcif 4463 〈cop 4563 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 1oc1o 8084 ≼ cdom 8495 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: unxpdom2 8714 sucxpdom 8715 djuxpdom 9599 |
Copyright terms: Public domain | W3C validator |