Theorem oaf1o 8601

Description: Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
oaf1o	⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem oaf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oacl 8573	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
2		oaword1 8590	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑥))
3		ontri1 6418	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐴 +o 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
4	1, 3	syldan 591	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
5	2, 4	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
6	1, 5	eldifd 3962	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
7	6	ralrimiva 3146	. 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
8		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9		eldifi 4131	. . . . . 6 ⊢ (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
10	9	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11		eldifn 4132	. . . . . . 7 ⊢ (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
12	11	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
13		ontri1 6418	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
14	10, 13	syldan 591	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
15	12, 14	mpbird 257	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ⊆ 𝑦)
16		oawordeu 8593	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝑦) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
17	8, 10, 15, 16	syl21anc 838	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
18		eqcom 2744	. . . . 5 ⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥))
19	18	reubii 3389	. . . 4 ⊢ (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
20	17, 19	sylib 218	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
21	20	ralrimiva 3146	. 2 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
22		eqid 2737	. . 3 ⊢ (𝑥 ∈ On ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +o 𝑥))
23	22	f1ompt 7131	. 2 ⊢ ((𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴) ↔ (∀𝑥 ∈ On (𝐴 +o 𝑥) ∈ (On ∖ 𝐴) ∧ ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥)))
24	7, 21, 23	sylanbrc 583	1 ⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378   ∖ cdif 3948   ⊆ wss 3951   ↦ cmpt 5225  Oncon0 6384  –1-1-onto→wf1o 6560  (class class class)co 7431   +_o coa 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510

This theorem is referenced by:  oacomf1olem  8602