Theorem domunsn 9167

Description: Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
domunsn	⊢ (𝐴 ≺ 𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Proof of Theorem domunsn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdom0 9148	. . . . 5 ⊢ ¬ 𝐴 ≺ ∅
2		breq2 5147	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ≺ 𝐵 ↔ 𝐴 ≺ ∅))
3	1, 2	mtbiri 327	. . . 4 ⊢ (𝐵 = ∅ → ¬ 𝐴 ≺ 𝐵)
4	3	con2i 139	. . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 = ∅)
5		neq0 4352	. . 3 ⊢ (¬ 𝐵 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝐵)
6	4, 5	sylib 218	. 2 ⊢ (𝐴 ≺ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
7		domdifsn 9094	. . . . 5 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝑧}))
8	7	adantr 480	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝑧}))
9		en2sn 9081	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝑧 ∈ V) → {𝐶} ≈ {𝑧})
10	9	elvd 3486	. . . . . 6 ⊢ (𝐶 ∈ V → {𝐶} ≈ {𝑧})
11		endom 9019	. . . . . 6 ⊢ ({𝐶} ≈ {𝑧} → {𝐶} ≼ {𝑧})
12	10, 11	syl 17	. . . . 5 ⊢ (𝐶 ∈ V → {𝐶} ≼ {𝑧})
13		snprc 4717	. . . . . . 7 ⊢ (¬ 𝐶 ∈ V ↔ {𝐶} = ∅)
14	13	biimpi 216	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → {𝐶} = ∅)
15		vsnex 5434	. . . . . . 7 ⊢ {𝑧} ∈ V
16	15	0dom 9146	. . . . . 6 ⊢ ∅ ≼ {𝑧}
17	14, 16	eqbrtrdi 5182	. . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐶} ≼ {𝑧})
18	12, 17	pm2.61i 182	. . . 4 ⊢ {𝐶} ≼ {𝑧}
19		disjdifr 4473	. . . . 5 ⊢ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅
20		undom 9099	. . . . 5 ⊢ (((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) ∧ ((𝐵 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
21	19, 20	mpan2 691	. . . 4 ⊢ ((𝐴 ≼ (𝐵 ∖ {𝑧}) ∧ {𝐶} ≼ {𝑧}) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
22	8, 18, 21	sylancl 586	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐴 ∪ {𝐶}) ≼ ((𝐵 ∖ {𝑧}) ∪ {𝑧}))
23		uncom 4158	. . . 4 ⊢ ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝐵 ∖ {𝑧}))
24		simpr 484	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
25	24	snssd 4809	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵)
26		undif 4482	. . . . 5 ⊢ ({𝑧} ⊆ 𝐵 ↔ ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
27	25, 26	sylib 218	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → ({𝑧} ∪ (𝐵 ∖ {𝑧})) = 𝐵)
28	23, 27	eqtrid 2789	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝐵 ∖ {𝑧}) ∪ {𝑧}) = 𝐵)
29	22, 28	breqtrd 5169	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐴 ∪ {𝐶}) ≼ 𝐵)
30	6, 29	exlimddv 1935	1 ⊢ (𝐴 ≺ 𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   ≈ cen 8982   ≼ cdom 8983   ≺ csdm 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986  df-dom 8987  df-sdom 8988

This theorem is referenced by:  canthp1lem1  10692