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Theorem omxpen 7947
 Description: The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
Assertion
Ref Expression
omxpen ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ≈ (𝐴 × 𝐵))

Proof of Theorem omxpen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcomeng 7937 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 xpexg 6858 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 × 𝐴) ∈ V)
32ancoms 468 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ∈ V)
4 omcl 7503 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
5 eqid 2610 . . . . 5 (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
65omxpenlem 7946 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))
7 f1oen2g 7858 . . . 4 (((𝐵 × 𝐴) ∈ V ∧ (𝐴 ·𝑜 𝐵) ∈ On ∧ (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵)) → (𝐵 × 𝐴) ≈ (𝐴 ·𝑜 𝐵))
83, 4, 6, 7syl3anc 1318 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ≈ (𝐴 ·𝑜 𝐵))
9 entr 7894 . . 3 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 ·𝑜 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 ·𝑜 𝐵))
101, 8, 9syl2anc 691 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐴 ·𝑜 𝐵))
1110ensymd 7893 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ≈ (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   × cxp 5036  Oncon0 5640  –1-1-onto→wf1o 5803  (class class class)co 6549   ↦ cmpt2 6551   +𝑜 coa 7444   ·𝑜 comu 7445   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-en 7842 This theorem is referenced by:  xpnum  8660  infxpenc2  8728
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