Theorem oaf1o 8578

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 8550	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
2		oaword1 8567	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑥))
3		ontri1 6400	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐴 +o 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
4	1, 3	syldan 603	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
5	2, 4	mpbid 235	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
6	1, 5	eldifd 4049	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
7	6	ralrimiva 3154	. 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
8		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9		eldifi 4066	. . . . . 6 ⊢ (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
10	9	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11		eldifn 4067	. . . . . . 7 ⊢ (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
12	11	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
13		ontri1 6400	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
14	10, 13	syldan 603	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
15	12, 14	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ⊆ 𝑦)
16		oawordeu 8570	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝑦) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
17	8, 10, 15, 16	syl21anc 851	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
18		eqcom 2767	. . . . 5 ⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥))
19	18	reubii 3374	. . . 4 ⊢ (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
20	17, 19	sylib 221	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
21	20	ralrimiva 3154	. 2 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
22		eqid 2760	. . 3 ⊢ (𝑥 ∈ On ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +o 𝑥))
23	22	f1ompt 7116	. 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 595	1 ⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363   ⊆ wss 3896   ∖ cdif 4044   ↦ cmpt 5185  Oncon0 6365  –1-1-onto→wf1o 6542  (class class class)co 7424   +_o coa 8480

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-oadd 8487

This theorem is used by:  oacomf1olem  8579