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Theorem domunsn 8317
Description: Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
domunsn (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Proof of Theorem domunsn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sdom0 8299 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 4813 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 318 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 136 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4094 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 209 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 8250 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 472 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 en2sn 8244 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
109elvd 3355 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
11 endom 8187 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1210, 11syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
13 snprc 4408 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 207 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
15 snex 5064 . . . . . . 7 {𝑧} ∈ V
16150dom 8297 . . . . . 6 ∅ ≼ {𝑧}
1714, 16syl6eqbr 4848 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1812, 17pm2.61i 176 . . . 4 {𝐶} ≼ {𝑧}
19 incom 3967 . . . . . 6 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝐵 ∖ {𝑧}))
20 disjdif 4200 . . . . . 6 ({𝑧} ∩ (𝐵 ∖ {𝑧})) = ∅
2119, 20eqtri 2787 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
22 undom 8255 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2321, 22mpan2 682 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
248, 18, 23sylancl 580 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
25 uncom 3919 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
26 simpr 477 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2726snssd 4494 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
28 undif 4209 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
2927, 28sylib 209 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3025, 29syl5eq 2811 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
3124, 30breqtrd 4835 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
326, 31exlimddv 2030 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334   class class class wbr 4809  cen 8157  cdom 8158  csdm 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-suc 5914  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163
This theorem is referenced by:  canthp1lem1  9727
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