Theorem omxpen 9073

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 9063	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		xpexg 7733	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ∈ V)
4		omcl 8534	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
5		eqid 2726	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
6	5	omxpenlem 9072	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)):(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
7		f1oen2g 8963	. . . 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 1368	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 × 𝐴) ≈ (𝐴 ·o 𝐵))
9		entr 9001	. . 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 9000	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ≈ (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141   × cxp 5667  Oncon0 6357  –1-1-onto→wf1o 6535  (class class class)co 7404   ∈ cmpo 7406   +_o coa 8461   ·_o comu 8462   ≈ cen 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-oadd 8468  df-omul 8469  df-er 8702  df-en 8939

This theorem is referenced by:  xpnum  9945  infxpenc2  10016