Theorem onadju 9619

Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Jim Kingdon, 7-Sep-2023.)

Assertion

Ref	Expression
onadju	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴 ⊔ 𝐵))

Proof of Theorem onadju

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrefg 8541	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
2	1	adantr 483	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ≈ 𝐴)
3		simpr 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4		eqid 2821	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8190	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
6	5	ancoms 461	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
7	6	simpld 497	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1oeng 8528	. . . . 5 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) → 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
9	3, 7, 8	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
10		incom 4178	. . . . 5 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 498	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	syl5eq 2868	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13		djuenun 9596	. . . 4 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∧ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
14	2, 9, 12, 13	syl3anc 1367	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
15		oarec 8188	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
16	14, 15	breqtrrd 5094	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵))
17	16	ensymd 8560	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3934   ∩ cin 3935  ∅c0 4291   class class class wbr 5066   ↦ cmpt 5146  ran crn 5556  Oncon0 6191  –1-1-onto→wf1o 6354  (class class class)co 7156   +_o coa 8099   ≈ cen 8506   ⊔ cdju 9327

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dju 9330

This theorem is referenced by:  cardadju  9620  nnadju  9623  tr3dom  39943