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Theorem unxpdom2 8702
Description: Corollary of unxpdom 8701. (Contributed by NM, 16-Sep-2004.)
Assertion
Ref Expression
unxpdom2 ((1o𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))

Proof of Theorem unxpdom2
StepHypRef Expression
1 relsdom 8491 . . . . . . . 8 Rel ≺
21brrelex2i 5582 . . . . . . 7 (1o𝐴𝐴 ∈ V)
32adantr 484 . . . . . 6 ((1o𝐴𝐵𝐴) → 𝐴 ∈ V)
4 1onn 8240 . . . . . 6 1o ∈ ω
5 xpsneng 8577 . . . . . 6 ((𝐴 ∈ V ∧ 1o ∈ ω) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 589 . . . . 5 ((1o𝐴𝐵𝐴) → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 8535 . . . 4 ((1o𝐴𝐵𝐴) → 𝐴 ≈ (𝐴 × {1o}))
8 endom 8511 . . . 4 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . 3 ((1o𝐴𝐵𝐴) → 𝐴 ≼ (𝐴 × {1o}))
10 simpr 488 . . . 4 ((1o𝐴𝐵𝐴) → 𝐵𝐴)
11 0ex 5184 . . . . . 6 ∅ ∈ V
12 xpsneng 8577 . . . . . 6 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
133, 11, 12sylancl 589 . . . . 5 ((1o𝐴𝐵𝐴) → (𝐴 × {∅}) ≈ 𝐴)
1413ensymd 8535 . . . 4 ((1o𝐴𝐵𝐴) → 𝐴 ≈ (𝐴 × {∅}))
15 domentr 8543 . . . 4 ((𝐵𝐴𝐴 ≈ (𝐴 × {∅})) → 𝐵 ≼ (𝐴 × {∅}))
1610, 14, 15syl2anc 587 . . 3 ((1o𝐴𝐵𝐴) → 𝐵 ≼ (𝐴 × {∅}))
17 1n0 8094 . . . 4 1o ≠ ∅
18 xpsndisj 5993 . . . 4 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
1917, 18mp1i 13 . . 3 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
20 undom 8580 . . 3 (((𝐴 ≼ (𝐴 × {1o}) ∧ 𝐵 ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴𝐵) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
219, 16, 19, 20syl21anc 836 . 2 ((1o𝐴𝐵𝐴) → (𝐴𝐵) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
22 sdomentr 8627 . . . . 5 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
237, 22syldan 594 . . . 4 ((1o𝐴𝐵𝐴) → 1o ≺ (𝐴 × {1o}))
24 sdomentr 8627 . . . . 5 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
2514, 24syldan 594 . . . 4 ((1o𝐴𝐵𝐴) → 1o ≺ (𝐴 × {∅}))
26 unxpdom 8701 . . . 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 8656 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
296, 13, 28syl2anc 587 . . 3 ((1o𝐴𝐵𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
30 domentr 8543 . . 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 8537 . 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 1538  wcel 2115  wne 3007  Vcvv 3471  cun 3908  cin 3909  c0 4266  {csn 4540   class class class wbr 5039   × cxp 5526  ωcom 7555  1oc1o 8070  cen 8481  cdom 8482  csdm 8483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-1st 7664  df-2nd 7665  df-1o 8077  df-2o 8078  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487
This theorem is referenced by: (None)
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