Theorem unxpdom 9233

Description: Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
unxpdom	⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))

Proof of Theorem unxpdom

Dummy variables 𝑥 𝑦 𝑢 𝑡 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 8963	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5716	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5716	. . 3 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V)
4	2, 3	anim12i 625	. 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		breq2 5111	. . . . 5 ⊢ (𝑥 = 𝐴 → (1o ≺ 𝑥 ↔ 1o ≺ 𝐴))
6	5	anbi1d 643	. . . 4 ⊢ (𝑥 = 𝐴 → ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝑦)))
7		uneq1 4111	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
8		xpeq1 5673	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
9	7, 8	breq12d 5120	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))
10	6, 9	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → (((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))))
11		breq2 5111	. . . . 5 ⊢ (𝑦 = 𝐵 → (1o ≺ 𝑦 ↔ 1o ≺ 𝐵))
12	11	anbi2d 642	. . . 4 ⊢ (𝑦 = 𝐵 → ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝐵)))
13		uneq2 4112	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
14		xpeq2 5680	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
15	13, 14	breq12d 5120	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))
16	12, 15	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐵 → (((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))))
17		eqid 2762	. . . 4 ⊢ (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩)) = (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩))
18		eqid 2762	. . . 4 ⊢ if(𝑧 ∈ 𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩) = if(𝑧 ∈ 𝑥, ⟨𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)⟩, ⟨if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧⟩)
19	17, 18	unxpdomlem3 9232	. . 3 ⊢ ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦))
20	10, 16, 19	vtocl2g 3536	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))
21	4, 20	mpcom 39	1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900  ifcif 4485  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   × cxp 5657  1_oc1o 8452   ≼ cdom 8954   ≺ csdm 8955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8459  df-2o 8460  df-en 8957  df-dom 8958  df-sdom 8959

This theorem is used by:  unxpdom2  9234  sucxpdom  9235  djuxpdom  10192