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Theorem domunsn 8277
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 8259 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 4808 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 316 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 4073 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 208 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 8210 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 472 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 vex 3343 . . . . . . 7 𝑧 ∈ V
10 en2sn 8204 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
119, 10mpan2 709 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
12 endom 8150 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1311, 12syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
14 snprc 4397 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1514biimpi 206 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
16 snex 5057 . . . . . . 7 {𝑧} ∈ V
17160dom 8257 . . . . . 6 ∅ ≼ {𝑧}
1815, 17syl6eqbr 4843 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1913, 18pm2.61i 176 . . . 4 {𝐶} ≼ {𝑧}
20 incom 3948 . . . . . 6 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝐵 ∖ {𝑧}))
21 disjdif 4184 . . . . . 6 ({𝑧} ∩ (𝐵 ∖ {𝑧})) = ∅
2220, 21eqtri 2782 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
23 undom 8215 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2422, 23mpan2 709 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
258, 19, 24sylancl 697 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
26 uncom 3900 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
27 simpr 479 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2827snssd 4485 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
29 undif 4193 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3028, 29sylib 208 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3126, 30syl5eq 2806 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
3225, 31breqtrd 4830 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
336, 32exlimddv 2012 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321   class class class wbr 4804  cen 8120  cdom 8121  csdm 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-1o 7730  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126
This theorem is referenced by:  canthp1lem1  9686
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