Theorem oaf1o 8575

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 8547	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ On)
2		oaword1 8564	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝑥))
3		ontri1 6386	. . . . . 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 3937	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
7	6	ralrimiva 3132	. 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +o 𝑥) ∈ (On ∖ 𝐴))
8		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On)
9		eldifi 4106	. . . . . 6 ⊢ (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On)
10	9	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On)
11		eldifn 4107	. . . . . . 7 ⊢ (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
12	11	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
13		ontri1 6386	. . . . . . 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 8567	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝑦) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
17	8, 10, 15, 16	syl21anc 837	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝑦)
18		eqcom 2742	. . . . 5 ⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥))
19	18	reubii 3368	. . . 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 3132	. 2 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +o 𝑥))
22		eqid 2735	. . 3 ⊢ (𝑥 ∈ On ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +o 𝑥))
23	22	f1ompt 7101	. 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 3051  ∃!wreu 3357   ∖ cdif 3923   ⊆ wss 3926   ↦ cmpt 5201  Oncon0 6352  –1-1-onto→wf1o 6530  (class class class)co 7405   +_o coa 8477

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-oadd 8484

This theorem is referenced by:  oacomf1olem  8576