Theorem undom 9054

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 5338. (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 4439	. 2 ⊢ (𝐴 ∪ (𝐶 ∖ 𝐴)) = (𝐴 ∪ 𝐶)
2		reldom 8950	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5720	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4	2	brrelex2i 5720	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → 𝐷 ∈ V)
5		unexg 7743	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
6	3, 4, 5	syl2an 607	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
7	6	adantr 485	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
8		brdomi 8957	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥 𝑥:𝐴–1-1→𝐵)
9		brdomi 8957	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → ∃𝑦 𝑦:𝐶–1-1→𝐷)
10		exdistrv 1985	. . . . . 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 4434	. . . . . . . . . 10 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅
12		difss 4091	. . . . . . . . . . . 12 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
13		f1ssres 6785	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐶–1-1→𝐷 ∧ (𝐶 ∖ 𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
14	12, 13	mpan2 703	. . . . . . . . . . 11 ⊢ (𝑦:𝐶–1-1→𝐷 → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
15		f1un 6843	. . . . . . . . . . 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 693	. . . . . . . . . 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 715	. . . . . . . . 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 3459	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
20	19	resex 6030	. . . . . . . . . . . 12 ⊢ (𝑦 ↾ (𝐶 ∖ 𝐴)) ∈ V
21	18, 20	unex 7744	. . . . . . . . . . 11 ⊢ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V
22		f1dom3g 8965	. . . . . . . . . . 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 1477	. . . . . . . . . 10 ⊢ (((𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
24	23	expcom 418	. . . . . . . . 9 ⊢ ((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
25	17, 24	syl 18	. . . . . . . 8 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
26	25	ex 417	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
27	26	exlimivv 1962	. . . . . 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 238	. . . . 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 607	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
30	29	imp 411	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
31	7, 30	mpd 16	. 2 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
32	1, 31	eqbrtrrid 5148	1 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287   class class class wbr 5110   ↾ cres 5665  –1-1→wf1 6535   ≼ cdom 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-dom 8946

This theorem is referenced by:  domunsncan  9066  domunsn  9116  sucdom2  9188  unxpdom2  9221  sucxpdom  9222  fodomfi  9273  undjudom  10152  djudom1  10167