Theorem unxpdom 8727

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 8518	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5611	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3	1	brrelex2i 5611	. . 3 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V)
4	2, 3	anim12i 614	. 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		breq2 5072	. . . . 5 ⊢ (𝑥 = 𝐴 → (1o ≺ 𝑥 ↔ 1o ≺ 𝐴))
6	5	anbi1d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝑦)))
7		uneq1 4134	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
8		xpeq1 5571	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
9	7, 8	breq12d 5081	. . . 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 5072	. . . . 5 ⊢ (𝑦 = 𝐵 → (1o ≺ 𝑦 ↔ 1o ≺ 𝐵))
12	11	anbi2d 630	. . . 4 ⊢ (𝑦 = 𝐵 → ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝐵)))
13		uneq2 4135	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
14		xpeq2 5578	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
15	13, 14	breq12d 5081	. . . 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 2823	. . . 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 2823	. . . 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 8726	. . 3 ⊢ ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦))
20	10, 16, 19	vtocl2g 3574	. 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 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∪ cun 3936  ifcif 4469  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  1_oc1o 8097   ≼ cdom 8509   ≺ csdm 8510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-1o 8104  df-2o 8105  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514

This theorem is referenced by:  unxpdom2  8728  sucxpdom  8729  djuxpdom  9613