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Theorem inf3lem6 8568
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8570 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 3234 . . . . . . . . . . 11 𝑢 ∈ V
4 vex 3234 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 8567 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → (𝐹𝑣) ⊊ (𝐹𝑢)))
6 dfpss2 3725 . . . . . . . . . . 11 ((𝐹𝑣) ⊊ (𝐹𝑢) ↔ ((𝐹𝑣) ⊆ (𝐹𝑢) ∧ ¬ (𝐹𝑣) = (𝐹𝑢)))
76simprbi 479 . . . . . . . . . 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 752 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
111, 2, 4, 3inf3lem5 8567 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → (𝐹𝑢) ⊊ (𝐹𝑣)))
12 dfpss2 3725 . . . . . . . . . . . 12 ((𝐹𝑢) ⊊ (𝐹𝑣) ↔ ((𝐹𝑢) ⊆ (𝐹𝑣) ∧ ¬ (𝐹𝑢) = (𝐹𝑣)))
1312simprbi 479 . . . . . . . . . . 11 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑢) = (𝐹𝑣))
14 eqcom 2658 . . . . . . . . . . 11 ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑣) = (𝐹𝑢))
1513, 14sylnib 317 . . . . . . . . . 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 753 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1910, 18jaod 394 . . . . . 6 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣𝑢𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
2019con2d 129 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → ¬ (𝑣𝑢𝑢𝑣)))
21 nnord 7115 . . . . . . 7 (𝑣 ∈ ω → Ord 𝑣)
22 nnord 7115 . . . . . . 7 (𝑢 ∈ ω → Ord 𝑢)
23 ordtri3 5797 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2421, 22, 23syl2an 493 . . . . . 6 ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2524adantl 481 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2620, 25sylibrd 249 . . . 4 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
2726ralrimivva 3000 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
28 frfnom 7575 . . . . . 6 (rec(𝐺, ∅) ↾ ω) Fn ω
29 fneq1 6017 . . . . . 6 (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
3028, 29mpbiri 248 . . . . 5 (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31 fvelrnb 6282 . . . . . . . 8 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹𝑣) = 𝑢))
32 inf3lem.4 . . . . . . . . . . . 12 𝐵 ∈ V
331, 2, 4, 32inf3lemd 8562 . . . . . . . . . . 11 (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥)
34 fvex 6239 . . . . . . . . . . . 12 (𝐹𝑣) ∈ V
3534elpw 4197 . . . . . . . . . . 11 ((𝐹𝑣) ∈ 𝒫 𝑥 ↔ (𝐹𝑣) ⊆ 𝑥)
3633, 35sylibr 224 . . . . . . . . . 10 (𝑣 ∈ ω → (𝐹𝑣) ∈ 𝒫 𝑥)
37 eleq1 2718 . . . . . . . . . 10 ((𝐹𝑣) = 𝑢 → ((𝐹𝑣) ∈ 𝒫 𝑥𝑢 ∈ 𝒫 𝑥))
3836, 37syl5ibcom 235 . . . . . . . . 9 (𝑣 ∈ ω → ((𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥))
3938rexlimiv 3056 . . . . . . . 8 (∃𝑣 ∈ ω (𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥)
4031, 39syl6bi 243 . . . . . . 7 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹𝑢 ∈ 𝒫 𝑥))
4140ssrdv 3642 . . . . . 6 (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
4241ancli 573 . . . . 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 5930 . . . 4 (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
4543, 44mpbir 221 . . 3 𝐹:ω⟶𝒫 𝑥
4627, 45jctil 559 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
47 dff13 6552 . 2 (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
4846, 47sylibr 224 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607  wpss 3608  c0 3948  𝒫 cpw 4191   cuni 4468  cmpt 4762  ran crn 5144  cres 5145  Ord word 5760   Fn wfn 5921  wf 5922  1-1wf1 5923  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  inf3lem7  8569  dominf  9305  dominfac  9433
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