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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 8969 | . . . . 5 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5734 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V) |
3 | endom 9000 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
4 | xpdom1g 9094 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
5 | 2, 3, 4 | syl2anr 595 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
6 | 1 | brrelex2i 5735 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | endom 9000 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≼ 𝐷) | |
8 | xpdom2g 9093 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ≼ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) | |
9 | 6, 7, 8 | syl2an 594 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) |
10 | domtr 9028 | . . 3 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) | |
11 | 5, 9, 10 | syl2anc 582 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) |
12 | 1 | brrelex2i 5735 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
13 | ensym 9024 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
14 | endom 9000 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
16 | xpdom1g 9094 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐵 ≼ 𝐴) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) | |
17 | 12, 15, 16 | syl2anr 595 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) |
18 | 1 | brrelex1i 5734 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
19 | ensym 9024 | . . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ 𝐶) | |
20 | endom 9000 | . . . . 5 ⊢ (𝐷 ≈ 𝐶 → 𝐷 ≼ 𝐶) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≼ 𝐶) |
22 | xpdom2g 9093 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐷 ≼ 𝐶) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) | |
23 | 18, 21, 22 | syl2an 594 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) |
24 | domtr 9028 | . . 3 ⊢ (((𝐵 × 𝐷) ≼ (𝐴 × 𝐷) ∧ (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) | |
25 | 17, 23, 24 | syl2anc 582 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) |
26 | sbth 9118 | . 2 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐷) ∧ (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | |
27 | 11, 25, 26 | syl2anc 582 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 class class class wbr 5149 × cxp 5676 ≈ cen 8961 ≼ cdom 8962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-1st 7994 df-2nd 7995 df-er 8725 df-en 8965 df-dom 8966 |
This theorem is referenced by: map2xp 9172 unxpdom2 9279 sucxpdom 9280 xpnum 9976 infxpenlem 10038 infxpidm2 10042 xpdjuen 10204 mapdjuen 10205 pwdjuen 10206 djuxpdom 10210 ackbij1lem5 10249 canthp1lem1 10677 xpnnen 16191 qnnen 16193 rexpen 16208 met2ndci 24475 re2ndc 24761 dyadmbl 25573 opnmblALT 25576 mbfimaopnlem 25628 mblfinlem1 37258 |
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