Theorem unxpdom 9171

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 8902	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5689	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5689	. . 3 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V)
4	2, 3	anim12i 614	. 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		breq2 5104	. . . . 5 ⊢ (𝑥 = 𝐴 → (1o ≺ 𝑥 ↔ 1o ≺ 𝐴))
6	5	anbi1d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝑦)))
7		uneq1 4115	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
8		xpeq1 5646	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
9	7, 8	breq12d 5113	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))
10	6, 9	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → (((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))))
11		breq2 5104	. . . . 5 ⊢ (𝑦 = 𝐵 → (1o ≺ 𝑦 ↔ 1o ≺ 𝐵))
12	11	anbi2d 631	. . . 4 ⊢ (𝑦 = 𝐵 → ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝐵)))
13		uneq2 4116	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
14		xpeq2 5653	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
15	13, 14	breq12d 5113	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))
16	12, 15	imbi12d 344	. . 3 ⊢ (𝑦 = 𝐵 → (((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))))
17		eqid 2737	. . . 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 2737	. . . 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 9170	. . 3 ⊢ ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦))
20	10, 16, 19	vtocl2g 3531	. 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 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901  ifcif 4481  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  1_oc1o 8400   ≼ cdom 8893   ≺ csdm 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8407  df-2o 8408  df-en 8896  df-dom 8897  df-sdom 8898

This theorem is referenced by:  unxpdom2  9172  sucxpdom  9173  djuxpdom  10108