Theorem omxpen 9140

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 9130	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		xpexg 7785	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ∈ V)
4		omcl 8592	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
5		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
6	5	omxpenlem 9139	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
7		f1oen2g 9028	. . . 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 1371	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵))
9		entr 9066	. . 3 ⊢ (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 ·o 𝐵))
10	1, 8, 9	syl2anc 583	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐴 ·o 𝐵))
11	10	ensymd 9065	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ≈ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   × cxp 5698  Oncon0 6395  –1-1-onto→wf1o 6572  (class class class)co 7448   ∈ cmpo 7450   +_o coa 8519   ·_o comu 8520   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-er 8763  df-en 9004

This theorem is referenced by:  xpnum  10020  infxpenc2  10091