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Theorem inf3lem6 9678
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9680 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 3466 . . . . . . . . . . 11 𝑢 ∈ V
4 vex 3466 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 9677 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → (𝐹𝑣) ⊊ (𝐹𝑢)))
6 dfpss2 4084 . . . . . . . . . . 11 ((𝐹𝑣) ⊊ (𝐹𝑢) ↔ ((𝐹𝑣) ⊆ (𝐹𝑢) ∧ ¬ (𝐹𝑣) = (𝐹𝑢)))
76simprbi 495 . . . . . . . . . 10 ((𝐹𝑣) ⊊ (𝐹𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢))
85, 7syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢)))
98expdimp 451 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑢 ∈ ω) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
109adantrl 714 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
111, 2, 4, 3inf3lem5 9677 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → (𝐹𝑢) ⊊ (𝐹𝑣)))
12 dfpss2 4084 . . . . . . . . . . . 12 ((𝐹𝑢) ⊊ (𝐹𝑣) ↔ ((𝐹𝑢) ⊆ (𝐹𝑣) ∧ ¬ (𝐹𝑢) = (𝐹𝑣)))
1312simprbi 495 . . . . . . . . . . 11 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑢) = (𝐹𝑣))
14 eqcom 2733 . . . . . . . . . . 11 ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑣) = (𝐹𝑢))
1513, 14sylnib 327 . . . . . . . . . 10 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢))
1611, 15syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
1716expdimp 451 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑣 ∈ ω) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1817adantrr 715 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1910, 18jaod 857 . . . . . 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 7886 . . . . . . 7 (𝑣 ∈ ω → Ord 𝑣)
22 nnord 7886 . . . . . . 7 (𝑢 ∈ ω → Ord 𝑢)
23 ordtri3 6414 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2421, 22, 23syl2an 594 . . . . . 6 ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2524adantl 480 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2620, 25sylibrd 258 . . . 4 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
2726ralrimivva 3191 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
28 frfnom 8467 . . . . . 6 (rec(𝐺, ∅) ↾ ω) Fn ω
29 fneq1 6653 . . . . . 6 (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
3028, 29mpbiri 257 . . . . 5 (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31 fvelrnb 6965 . . . . . . . 8 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹𝑣) = 𝑢))
32 inf3lem.4 . . . . . . . . . . . 12 𝐵 ∈ V
331, 2, 4, 32inf3lemd 9672 . . . . . . . . . . 11 (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥)
34 fvex 6916 . . . . . . . . . . . 12 (𝐹𝑣) ∈ V
3534elpw 4611 . . . . . . . . . . 11 ((𝐹𝑣) ∈ 𝒫 𝑥 ↔ (𝐹𝑣) ⊆ 𝑥)
3633, 35sylibr 233 . . . . . . . . . 10 (𝑣 ∈ ω → (𝐹𝑣) ∈ 𝒫 𝑥)
37 eleq1 2814 . . . . . . . . . 10 ((𝐹𝑣) = 𝑢 → ((𝐹𝑣) ∈ 𝒫 𝑥𝑢 ∈ 𝒫 𝑥))
3836, 37syl5ibcom 244 . . . . . . . . 9 (𝑣 ∈ ω → ((𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥))
3938rexlimiv 3138 . . . . . . . 8 (∃𝑣 ∈ ω (𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥)
4031, 39biimtrdi 252 . . . . . . 7 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹𝑢 ∈ 𝒫 𝑥))
4140ssrdv 3985 . . . . . 6 (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
4241ancli 547 . . . . 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 6560 . . . 4 (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
4543, 44mpbir 230 . . 3 𝐹:ω⟶𝒫 𝑥
4627, 45jctil 518 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
47 dff13 7272 . 2 (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
4846, 47sylibr 233 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  cin 3946  wss 3947  wpss 3948  c0 4325  𝒫 cpw 4607   cuni 4915  cmpt 5238  ran crn 5685  cres 5686  Ord word 6377   Fn wfn 6551  wf 6552  1-1wf1 6553  cfv 6556  ωcom 7878  reccrdg 8441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748  ax-reg 9637
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7429  df-om 7879  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442
This theorem is referenced by:  inf3lem7  9679  dominf  10490  dominfac  10618
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