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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 8517 | . . . . 5 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5611 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V) |
3 | endom 8539 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
4 | xpdom1g 8617 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
5 | 2, 3, 4 | syl2anr 598 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
6 | 1 | brrelex2i 5612 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | endom 8539 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≼ 𝐷) | |
8 | xpdom2g 8616 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ≼ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) | |
9 | 6, 7, 8 | syl2an 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) |
10 | domtr 8565 | . . 3 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) | |
11 | 5, 9, 10 | syl2anc 586 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) |
12 | 1 | brrelex2i 5612 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
13 | ensym 8561 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
14 | endom 8539 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
16 | xpdom1g 8617 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐵 ≼ 𝐴) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) | |
17 | 12, 15, 16 | syl2anr 598 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) |
18 | 1 | brrelex1i 5611 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
19 | ensym 8561 | . . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ 𝐶) | |
20 | endom 8539 | . . . . 5 ⊢ (𝐷 ≈ 𝐶 → 𝐷 ≼ 𝐶) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≼ 𝐶) |
22 | xpdom2g 8616 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐷 ≼ 𝐶) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) | |
23 | 18, 21, 22 | syl2an 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) |
24 | domtr 8565 | . . 3 ⊢ (((𝐵 × 𝐷) ≼ (𝐴 × 𝐷) ∧ (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) | |
25 | 17, 23, 24 | syl2anc 586 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) |
26 | sbth 8640 | . 2 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐷) ∧ (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | |
27 | 11, 25, 26 | syl2anc 586 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 × cxp 5556 ≈ cen 8509 ≼ cdom 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 |
This theorem is referenced by: map2xp 8690 unxpdom2 8729 sucxpdom 8730 xpnum 9383 infxpenlem 9442 infxpidm2 9446 xpdjuen 9608 mapdjuen 9609 pwdjuen 9610 djuxpdom 9614 ackbij1lem5 9649 canthp1lem1 10077 xpnnen 15567 qnnen 15569 rexpen 15584 met2ndci 23135 re2ndc 23412 dyadmbl 24204 opnmblALT 24207 mbfimaopnlem 24259 mblfinlem1 34933 |
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