Theorem onadju 10263

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 9044	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ≈ 𝐴)
3		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4		eqid 2740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8620	. . . . . . 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 458	. . . . . 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 494	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1oeng 9031	. . . . 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 583	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
10		incom 4230	. . . . 5 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 495	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	eqtrid 2792	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13		djuenun 10240	. . . 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 1371	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
15		oarec 8618	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
16	14, 15	breqtrrd 5194	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵))
17	16	ensymd 9065	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴 ⊔ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ∩ cin 3975  ∅c0 4352   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701  Oncon0 6395  –1-1-onto→wf1o 6572  (class class class)co 7448   +_o coa 8519   ≈ cen 9000   ⊔ cdju 9967

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-er 8763  df-en 9004  df-dju 9970

This theorem is referenced by:  cardadju  10264  nnadjuALT  10268  tr3dom  43490