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Mirrors > Home > MPE Home > Th. List > xpen | Structured version Visualization version GIF version |
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
xpen | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8989 | . . . . 5 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5745 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V) |
3 | endom 9018 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
4 | xpdom1g 9108 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
5 | 2, 3, 4 | syl2anr 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
6 | 1 | brrelex2i 5746 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | endom 9018 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≼ 𝐷) | |
8 | xpdom2g 9107 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ≼ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) | |
9 | 6, 7, 8 | syl2an 596 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) |
10 | domtr 9046 | . . 3 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) | |
11 | 5, 9, 10 | syl2anc 584 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) |
12 | 1 | brrelex2i 5746 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
13 | ensym 9042 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
14 | endom 9018 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
16 | xpdom1g 9108 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐵 ≼ 𝐴) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) | |
17 | 12, 15, 16 | syl2anr 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) |
18 | 1 | brrelex1i 5745 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
19 | ensym 9042 | . . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ 𝐶) | |
20 | endom 9018 | . . . . 5 ⊢ (𝐷 ≈ 𝐶 → 𝐷 ≼ 𝐶) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≼ 𝐶) |
22 | xpdom2g 9107 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐷 ≼ 𝐶) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) | |
23 | 18, 21, 22 | syl2an 596 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) |
24 | domtr 9046 | . . 3 ⊢ (((𝐵 × 𝐷) ≼ (𝐴 × 𝐷) ∧ (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) | |
25 | 17, 23, 24 | syl2anc 584 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) |
26 | sbth 9132 | . 2 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐷) ∧ (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | |
27 | 11, 25, 26 | syl2anc 584 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 × cxp 5687 ≈ cen 8981 ≼ cdom 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 |
This theorem is referenced by: map2xp 9186 unxpdom2 9288 sucxpdom 9289 xpnum 9989 infxpenlem 10051 infxpidm2 10055 xpdjuen 10218 mapdjuen 10219 pwdjuen 10220 djuxpdom 10224 ackbij1lem5 10261 canthp1lem1 10690 xpnnen 16244 qnnen 16246 rexpen 16261 met2ndci 24551 re2ndc 24837 dyadmbl 25649 opnmblALT 25652 mbfimaopnlem 25704 mblfinlem1 37644 |
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