Theorem aomclem1 43482

Description: Lemma for dfac11 43490. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of (𝑅₁‘𝐴). In what follows, 𝐴 is the index of the rank we wish to well-order, 𝑧 is the collection of well-orderings constructed so far, dom 𝑧 is the set of ordinal indices of constructed ranks i.e. the next rank to construct, and 𝑦 is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
aomclem1.b	⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
aomclem1.on	⊢ (𝜑 → dom 𝑧 ∈ On)
aomclem1.su	⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
aomclem1.we	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))

Assertion

Ref	Expression
aomclem1	⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))

Distinct variable group:   𝑧,𝑎,𝑏,𝑐,𝑑

Allowed substitution hints:   𝜑(𝑧,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑧,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aomclem1

Step	Hyp	Ref	Expression
1		fvex 6853	. . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V
2		vex 3433	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	2	dmex 7860	. . . . . . 7 ⊢ dom 𝑧 ∈ V
4	3	uniex 7695	. . . . . 6 ⊢ ∪ dom 𝑧 ∈ V
5	4	sucid 6407	. . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧
6		aomclem1.su	. . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
7	5, 6	eleqtrrid 2843	. . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧)
8		aomclem1.we	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
9		fveq2 6840	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧))
10		fveq2 6840	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧))
11	9, 10	weeq12d 5620	. . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)))
12	11	rspcva 3562	. . . 4 ⊢ ((∪ dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))
13	7, 8, 12	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))
14		aomclem1.b	. . . 4 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
15	14	wepwso 43471	. . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
16	1, 13, 15	sylancr 588	. 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
17	6	fveq2d 6844	. . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧))
18		aomclem1.on	. . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On)
19		onuni 7742	. . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On)
20		r1suc 9694	. . . . 5 ⊢ (∪ dom 𝑧 ∈ On → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
21	18, 19, 20	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
22	17, 21	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
23		soeq2 5561	. . 3 ⊢ ((𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧) → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
24	22, 23	syl 17	. 2 ⊢ (𝜑 → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
25	16, 24	mpbird 257	1 ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085  {copab 5147   Or wor 5538   We wwe 5583  dom cdm 5631  Oncon0 6323  suc csuc 6325  ‘cfv 6498  𝑅₁cr1 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-map 8775  df-r1 9688

This theorem is referenced by:  aomclem2  43483