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Theorem inf3lem6 8390
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8392 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 3175 . . . . . . . . . . 11 𝑢 ∈ V
4 vex 3175 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 8389 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → (𝐹𝑣) ⊊ (𝐹𝑢)))
6 dfpss2 3653 . . . . . . . . . . 11 ((𝐹𝑣) ⊊ (𝐹𝑢) ↔ ((𝐹𝑣) ⊆ (𝐹𝑢) ∧ ¬ (𝐹𝑣) = (𝐹𝑢)))
76simprbi 478 . . . . . . . . . 10 ((𝐹𝑣) ⊊ (𝐹𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢))
85, 7syl6 34 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢)))
98expdimp 451 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑢 ∈ ω) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
109adantrl 747 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
111, 2, 4, 3inf3lem5 8389 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → (𝐹𝑢) ⊊ (𝐹𝑣)))
12 dfpss2 3653 . . . . . . . . . . . 12 ((𝐹𝑢) ⊊ (𝐹𝑣) ↔ ((𝐹𝑢) ⊆ (𝐹𝑣) ∧ ¬ (𝐹𝑢) = (𝐹𝑣)))
1312simprbi 478 . . . . . . . . . . 11 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑢) = (𝐹𝑣))
14 eqcom 2616 . . . . . . . . . . 11 ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑣) = (𝐹𝑢))
1513, 14sylnib 316 . . . . . . . . . 10 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢))
1611, 15syl6 34 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
1716expdimp 451 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑣 ∈ ω) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1817adantrr 748 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1910, 18jaod 393 . . . . . 6 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣𝑢𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
2019con2d 127 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → ¬ (𝑣𝑢𝑢𝑣)))
21 nnord 6942 . . . . . . 7 (𝑣 ∈ ω → Ord 𝑣)
22 nnord 6942 . . . . . . 7 (𝑢 ∈ ω → Ord 𝑢)
23 ordtri3 5661 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2421, 22, 23syl2an 492 . . . . . 6 ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2524adantl 480 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2620, 25sylibrd 247 . . . 4 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
2726ralrimivva 2953 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
28 frfnom 7394 . . . . . 6 (rec(𝐺, ∅) ↾ ω) Fn ω
29 fneq1 5878 . . . . . 6 (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
3028, 29mpbiri 246 . . . . 5 (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31 fvelrnb 6137 . . . . . . . 8 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹𝑣) = 𝑢))
32 inf3lem.4 . . . . . . . . . . . 12 𝐵 ∈ V
331, 2, 4, 32inf3lemd 8384 . . . . . . . . . . 11 (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥)
34 fvex 6097 . . . . . . . . . . . 12 (𝐹𝑣) ∈ V
3534elpw 4113 . . . . . . . . . . 11 ((𝐹𝑣) ∈ 𝒫 𝑥 ↔ (𝐹𝑣) ⊆ 𝑥)
3633, 35sylibr 222 . . . . . . . . . 10 (𝑣 ∈ ω → (𝐹𝑣) ∈ 𝒫 𝑥)
37 eleq1 2675 . . . . . . . . . 10 ((𝐹𝑣) = 𝑢 → ((𝐹𝑣) ∈ 𝒫 𝑥𝑢 ∈ 𝒫 𝑥))
3836, 37syl5ibcom 233 . . . . . . . . 9 (𝑣 ∈ ω → ((𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥))
3938rexlimiv 3008 . . . . . . . 8 (∃𝑣 ∈ ω (𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥)
4031, 39syl6bi 241 . . . . . . 7 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹𝑢 ∈ 𝒫 𝑥))
4140ssrdv 3573 . . . . . 6 (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
4241ancli 571 . . . . 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 5793 . . . 4 (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
4543, 44mpbir 219 . . 3 𝐹:ω⟶𝒫 𝑥
4627, 45jctil 557 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
47 dff13 6393 . 2 (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
4846, 47sylibr 222 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  cin 3538  wss 3539  wpss 3540  c0 3873  𝒫 cpw 4107   cuni 4366  cmpt 4637  ran crn 5028  cres 5029  Ord word 5624   Fn wfn 5784  wf 5785  1-1wf1 5786  cfv 5789  ωcom 6934  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  inf3lem7  8391  dominf  9127  dominfac  9251
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