Theorem undom 9030

Description: Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5319. (Revised by BTernaryTau, 4-Dec-2024.)

Assertion

Ref	Expression
undom	⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Proof of Theorem undom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		undif2 4428	. 2 ⊢ (𝐴 ∪ (𝐶 ∖ 𝐴)) = (𝐴 ∪ 𝐶)
2		reldom 8926	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5700	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4	2	brrelex2i 5700	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → 𝐷 ∈ V)
5		unexg 7720	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
6	3, 4, 5	syl2an 605	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
7	6	adantr 484	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
8		brdomi 8933	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥 𝑥:𝐴–1-1→𝐵)
9		brdomi 8933	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → ∃𝑦 𝑦:𝐶–1-1→𝐷)
10		exdistrv 1974	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ↔ (∃𝑥 𝑥:𝐴–1-1→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1→𝐷))
11		disjdif 4423	. . . . . . . . . 10 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅
12		difss 4087	. . . . . . . . . . . 12 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
13		f1ssres 6763	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐶–1-1→𝐷 ∧ (𝐶 ∖ 𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
14	12, 13	mpan2 701	. . . . . . . . . . 11 ⊢ (𝑦:𝐶–1-1→𝐷 → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
15		f1un 6821	. . . . . . . . . . 11 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
16	14, 15	sylanl2 691	. . . . . . . . . 10 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
17	11, 16	mpanr1 713	. . . . . . . . 9 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
18		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
20	19	resex 6011	. . . . . . . . . . . 12 ⊢ (𝑦 ↾ (𝐶 ∖ 𝐴)) ∈ V
21	18, 20	unex 7721	. . . . . . . . . . 11 ⊢ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V
22		f1dom3g 8941	. . . . . . . . . . 11 ⊢ (((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V ∧ (𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
23	21, 22	mp3an1 1468	. . . . . . . . . 10 ⊢ (((𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
24	23	expcom 417	. . . . . . . . 9 ⊢ ((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
25	17, 24	syl 17	. . . . . . . 8 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
26	25	ex 416	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
27	26	exlimivv 1951	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
28	10, 27	sylbir 237	. . . . 5 ⊢ ((∃𝑥 𝑥:𝐴–1-1→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
29	8, 9, 28	syl2an 605	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
30	29	imp 410	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
31	7, 30	mpd 15	. 2 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
32	1, 31	eqbrtrrid 5133	1 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283   class class class wbr 5097   ↾ cres 5645  –1-1→wf1 6512   ≼ cdom 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-dom 8922

This theorem is referenced by:  domunsncan  9042  domunsn  9092  sucdom2  9164  unxpdom2  9197  sucxpdom  9198  fodomfi  9249  undjudom  10117  djudom1  10132