Theorem omxpen 9113

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.

Step	Hyp	Ref	Expression
1		xpcomeng 9103	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		xpexg 7769	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ∈ V)
4		omcl 8573	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
5		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
6	5	omxpenlem 9112	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
7		f1oen2g 9008	. . . 4 ⊢ (((𝐵 × 𝐴) ∈ V ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵)) → (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵))
8	3, 4, 6, 7	syl3anc 1370	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵))
9		entr 9045	. . 3 ⊢ (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 ·o 𝐵))
10	1, 8, 9	syl2anc 584	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐴 ·o 𝐵))
11	10	ensymd 9044	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ≈ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148   × cxp 5687  Oncon0 6386  –1-1-onto→wf1o 6562  (class class class)co 7431   ∈ cmpo 7433   +_o coa 8502   ·_o comu 8503   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-en 8985

This theorem is referenced by:  xpnum  9989  infxpenc2  10060