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Theorem unxpdom2 8891
Description: Corollary of unxpdom 8890. (Contributed by NM, 16-Sep-2004.)
Assertion
Ref Expression
unxpdom2 ((1o𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))

Proof of Theorem unxpdom2
StepHypRef Expression
1 relsdom 8638 . . . . . . . 8 Rel ≺
21brrelex2i 5611 . . . . . . 7 (1o𝐴𝐴 ∈ V)
32adantr 484 . . . . . 6 ((1o𝐴𝐵𝐴) → 𝐴 ∈ V)
4 1onn 8372 . . . . . 6 1o ∈ ω
5 xpsneng 8735 . . . . . 6 ((𝐴 ∈ V ∧ 1o ∈ ω) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 589 . . . . 5 ((1o𝐴𝐵𝐴) → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 8684 . . . 4 ((1o𝐴𝐵𝐴) → 𝐴 ≈ (𝐴 × {1o}))
8 endom 8660 . . . 4 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . 3 ((1o𝐴𝐵𝐴) → 𝐴 ≼ (𝐴 × {1o}))
10 simpr 488 . . . 4 ((1o𝐴𝐵𝐴) → 𝐵𝐴)
11 0ex 5205 . . . . . 6 ∅ ∈ V
12 xpsneng 8735 . . . . . 6 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
133, 11, 12sylancl 589 . . . . 5 ((1o𝐴𝐵𝐴) → (𝐴 × {∅}) ≈ 𝐴)
1413ensymd 8684 . . . 4 ((1o𝐴𝐵𝐴) → 𝐴 ≈ (𝐴 × {∅}))
15 domentr 8692 . . . 4 ((𝐵𝐴𝐴 ≈ (𝐴 × {∅})) → 𝐵 ≼ (𝐴 × {∅}))
1610, 14, 15syl2anc 587 . . 3 ((1o𝐴𝐵𝐴) → 𝐵 ≼ (𝐴 × {∅}))
17 1n0 8226 . . . 4 1o ≠ ∅
18 xpsndisj 6031 . . . 4 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
1917, 18mp1i 13 . . 3 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
20 undom 8738 . . 3 (((𝐴 ≼ (𝐴 × {1o}) ∧ 𝐵 ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴𝐵) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
219, 16, 19, 20syl21anc 838 . 2 ((1o𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
22 sdomentr 8785 . . . . 5 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
237, 22syldan 594 . . . 4 ((1o𝐴𝐵𝐴) → 1o ≺ (𝐴 × {1o}))
24 sdomentr 8785 . . . . 5 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
2514, 24syldan 594 . . . 4 ((1o𝐴𝐵𝐴) → 1o ≺ (𝐴 × {∅}))
26 unxpdom 8890 . . . 4 ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
2723, 25, 26syl2anc 587 . . 3 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
28 xpen 8814 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
296, 13, 28syl2anc 587 . . 3 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
30 domentr 8692 . . 3 ((((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ (𝐴 × 𝐴))
3127, 29, 30syl2anc 587 . 2 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ (𝐴 × 𝐴))
32 domtr 8686 . 2 (((𝐴𝐵) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ (𝐴 × 𝐴)) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
3321, 31, 32syl2anc 587 1 ((1o𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  Vcvv 3413  cun 3869  cin 3870  c0 4242  {csn 4546   class class class wbr 5058   × cxp 5554  ωcom 7649  1oc1o 8200  cen 8628  cdom 8629  csdm 8630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-tp 4551  df-op 4553  df-uni 4825  df-int 4865  df-br 5059  df-opab 5121  df-mpt 5141  df-tr 5167  df-id 5460  df-eprel 5465  df-po 5473  df-so 5474  df-fr 5514  df-we 5516  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-ord 6221  df-on 6222  df-lim 6223  df-suc 6224  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-om 7650  df-1st 7766  df-2nd 7767  df-1o 8207  df-2o 8208  df-er 8396  df-en 8632  df-dom 8633  df-sdom 8634
This theorem is referenced by: (None)
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