Theorem onacda 9019

Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
onacda	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ≈ (𝐴 +𝑐 𝐵))

Proof of Theorem onacda

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrefg 7987	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
2	1	adantr 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ≈ 𝐴)
3		simpr 477	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
5	4	oacomf1olem 7644	. . . . . . 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 469	. . . . . 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 475	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
8		f1oeng 7974	. . . . 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 693	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
10		incom 3805	. . . . 5 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴)
11	6	simprd 479	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	syl5eq 2668	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅)
13		cdaenun 8996	. . . 4 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
14	2, 9, 12, 13	syl3anc 1326	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
15		oarec 7642	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
16	14, 15	breqtrrd 4681	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑐 𝐵) ≈ (𝐴 +𝑜 𝐵))
17	16	ensymd 8007	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ≈ (𝐴 +𝑐 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ∩ cin 3573  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  Oncon0 5723  –1-1-onto→wf1o 5887  (class class class)co 6650   +_𝑜 coa 7557   ≈ cen 7952   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  cardacda  9020  nnacda  9023