Theorem unxpdom 8208

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 8004	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5193	. . 3 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5193	. . 3 ⊢ (1𝑜 ≺ 𝐵 → 𝐵 ∈ V)
4	2, 3	anim12i 589	. 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		breq2 4689	. . . . 5 ⊢ (𝑥 = 𝐴 → (1𝑜 ≺ 𝑥 ↔ 1𝑜 ≺ 𝐴))
6	5	anbi1d 741	. . . 4 ⊢ (𝑥 = 𝐴 → ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦)))
7		uneq1 3793	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
8		xpeq1 5157	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
9	7, 8	breq12d 4698	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))
10	6, 9	imbi12d 333	. . 3 ⊢ (𝑥 = 𝐴 → (((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))))
11		breq2 4689	. . . . 5 ⊢ (𝑦 = 𝐵 → (1𝑜 ≺ 𝑦 ↔ 1𝑜 ≺ 𝐵))
12	11	anbi2d 740	. . . 4 ⊢ (𝑦 = 𝐵 → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵)))
13		uneq2 3794	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
14		xpeq2 5163	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
15	13, 14	breq12d 4698	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))
16	12, 15	imbi12d 333	. . 3 ⊢ (𝑦 = 𝐵 → (((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))))
17		eqid 2651	. . . 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 2651	. . . 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 8207	. . 3 ⊢ ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦))
20	10, 16, 19	vtocl2g 3301	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))
21	4, 20	mpcom 38	1 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605  ifcif 4119  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  1_𝑜c1o 7598   ≼ cdom 7995   ≺ csdm 7996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000

This theorem is referenced by:  unxpdom2  8209  sucxpdom  8210  cdaxpdom  9049