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Theorem inf3lem6 9673
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9675 for detailed description. (Contributed by NM, 29-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐵 ∈ V
Assertion
Ref Expression
inf3lem6 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Distinct variable group:   𝑥,𝑦,𝑤
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem inf3lem6
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inf3lem.1 . . . . . . . . . . 11 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
2 inf3lem.2 . . . . . . . . . . 11 𝐹 = (rec(𝐺, ∅) ↾ ω)
3 vex 3484 . . . . . . . . . . 11 𝑢 ∈ V
4 vex 3484 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 9672 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → (𝐹𝑣) ⊊ (𝐹𝑢)))
6 dfpss2 4088 . . . . . . . . . . 11 ((𝐹𝑣) ⊊ (𝐹𝑢) ↔ ((𝐹𝑣) ⊆ (𝐹𝑢) ∧ ¬ (𝐹𝑣) = (𝐹𝑢)))
76simprbi 496 . . . . . . . . . 10 ((𝐹𝑣) ⊊ (𝐹𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢))
85, 7syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢)))
98expdimp 452 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑢 ∈ ω) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
109adantrl 716 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
111, 2, 4, 3inf3lem5 9672 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → (𝐹𝑢) ⊊ (𝐹𝑣)))
12 dfpss2 4088 . . . . . . . . . . . 12 ((𝐹𝑢) ⊊ (𝐹𝑣) ↔ ((𝐹𝑢) ⊆ (𝐹𝑣) ∧ ¬ (𝐹𝑢) = (𝐹𝑣)))
1312simprbi 496 . . . . . . . . . . 11 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑢) = (𝐹𝑣))
14 eqcom 2744 . . . . . . . . . . 11 ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑣) = (𝐹𝑢))
1513, 14sylnib 328 . . . . . . . . . 10 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢))
1611, 15syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
1716expdimp 452 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑣 ∈ ω) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1817adantrr 717 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1910, 18jaod 860 . . . . . 6 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣𝑢𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
2019con2d 134 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → ¬ (𝑣𝑢𝑢𝑣)))
21 nnord 7895 . . . . . . 7 (𝑣 ∈ ω → Ord 𝑣)
22 nnord 7895 . . . . . . 7 (𝑢 ∈ ω → Ord 𝑢)
23 ordtri3 6420 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2421, 22, 23syl2an 596 . . . . . 6 ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2524adantl 481 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2620, 25sylibrd 259 . . . 4 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
2726ralrimivva 3202 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
28 frfnom 8475 . . . . . 6 (rec(𝐺, ∅) ↾ ω) Fn ω
29 fneq1 6659 . . . . . 6 (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
3028, 29mpbiri 258 . . . . 5 (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹𝑣) = 𝑢))
32 inf3lem.4 . . . . . . . . . . . 12 𝐵 ∈ V
331, 2, 4, 32inf3lemd 9667 . . . . . . . . . . 11 (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥)
34 fvex 6919 . . . . . . . . . . . 12 (𝐹𝑣) ∈ V
3534elpw 4604 . . . . . . . . . . 11 ((𝐹𝑣) ∈ 𝒫 𝑥 ↔ (𝐹𝑣) ⊆ 𝑥)
3633, 35sylibr 234 . . . . . . . . . 10 (𝑣 ∈ ω → (𝐹𝑣) ∈ 𝒫 𝑥)
37 eleq1 2829 . . . . . . . . . 10 ((𝐹𝑣) = 𝑢 → ((𝐹𝑣) ∈ 𝒫 𝑥𝑢 ∈ 𝒫 𝑥))
3836, 37syl5ibcom 245 . . . . . . . . 9 (𝑣 ∈ ω → ((𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥))
3938rexlimiv 3148 . . . . . . . 8 (∃𝑣 ∈ ω (𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥)
4031, 39biimtrdi 253 . . . . . . 7 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹𝑢 ∈ 𝒫 𝑥))
4140ssrdv 3989 . . . . . 6 (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
4241ancli 548 . . . . 5 (𝐹 Fn ω → (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
432, 30, 42mp2b 10 . . . 4 (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥)
44 df-f 6565 . . . 4 (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
4543, 44mpbir 231 . . 3 𝐹:ω⟶𝒫 𝑥
4627, 45jctil 519 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
47 dff13 7275 . 2 (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
4846, 47sylibr 234 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  wpss 3952  c0 4333  𝒫 cpw 4600   cuni 4907  cmpt 5225  ran crn 5686  cres 5687  Ord word 6383   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  ωcom 7887  reccrdg 8449
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-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
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-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-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-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  inf3lem7  9674  dominf  10485  dominfac  10613
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