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Theorem domunsn 8054
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 8036 . . . . 5 ¬ 𝐴 ≺ ∅
2 breq2 4617 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ≺ ∅))
31, 2mtbiri 317 . . . 4 (𝐵 = ∅ → ¬ 𝐴𝐵)
43con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐵 = ∅)
5 neq0 3906 . . 3 𝐵 = ∅ ↔ ∃𝑧 𝑧𝐵)
64, 5sylib 208 . 2 (𝐴𝐵 → ∃𝑧 𝑧𝐵)
7 domdifsn 7987 . . . . 5 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝑧}))
87adantr 481 . . . 4 ((𝐴𝐵𝑧𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9 vex 3189 . . . . . . 7 𝑧 ∈ V
10 en2sn 7981 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
119, 10mpan2 706 . . . . . 6 (𝐶 ∈ V → {𝐶} ≈ {𝑧})
12 endom 7926 . . . . . 6 ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
1311, 12syl 17 . . . . 5 (𝐶 ∈ V → {𝐶} ≼ {𝑧})
14 snprc 4223 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1514biimpi 206 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
16 snex 4869 . . . . . . 7 {𝑧} ∈ V
17160dom 8034 . . . . . 6 ∅ ≼ {𝑧}
1815, 17syl6eqbr 4652 . . . . 5 𝐶 ∈ V → {𝐶} ≼ {𝑧})
1913, 18pm2.61i 176 . . . 4 {𝐶} ≼ {𝑧}
20 incom 3783 . . . . . 6 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝐵 ∖ {𝑧}))
21 disjdif 4012 . . . . . 6 ({𝑧} ∩ (𝐵 ∖ {𝑧})) = ∅
2220, 21eqtri 2643 . . . . 5 ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
23 undom 7992 . . . . 5 (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
2422, 23mpan2 706 . . . 4 ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
258, 19, 24sylancl 693 . . 3 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
26 uncom 3735 . . . 4 ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
27 simpr 477 . . . . . 6 ((𝐴𝐵𝑧𝐵) → 𝑧𝐵)
2827snssd 4309 . . . . 5 ((𝐴𝐵𝑧𝐵) → {𝑧} ⊆ 𝐵)
29 undif 4021 . . . . 5 ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3028, 29sylib 208 . . . 4 ((𝐴𝐵𝑧𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
3126, 30syl5eq 2667 . . 3 ((𝐴𝐵𝑧𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
3225, 31breqtrd 4639 . 2 ((𝐴𝐵𝑧𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
336, 32exlimddv 1860 1 (𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148   class class class wbr 4613  cen 7896  cdom 7897  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  canthp1lem1  9418
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