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Theorem inf3lem6 8745
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8747 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 3353 . . . . . . . . . . 11 𝑢 ∈ V
4 vex 3353 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 8744 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → (𝐹𝑣) ⊊ (𝐹𝑢)))
6 dfpss2 3853 . . . . . . . . . . 11 ((𝐹𝑣) ⊊ (𝐹𝑢) ↔ ((𝐹𝑣) ⊆ (𝐹𝑢) ∧ ¬ (𝐹𝑣) = (𝐹𝑢)))
76simprbi 490 . . . . . . . . . 10 ((𝐹𝑣) ⊊ (𝐹𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢))
85, 7syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑢 ∈ ω ∧ 𝑣𝑢) → ¬ (𝐹𝑣) = (𝐹𝑢)))
98expdimp 444 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑢 ∈ ω) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
109adantrl 707 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣𝑢 → ¬ (𝐹𝑣) = (𝐹𝑢)))
111, 2, 4, 3inf3lem5 8744 . . . . . . . . . 10 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → (𝐹𝑢) ⊊ (𝐹𝑣)))
12 dfpss2 3853 . . . . . . . . . . . 12 ((𝐹𝑢) ⊊ (𝐹𝑣) ↔ ((𝐹𝑢) ⊆ (𝐹𝑣) ∧ ¬ (𝐹𝑢) = (𝐹𝑣)))
1312simprbi 490 . . . . . . . . . . 11 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑢) = (𝐹𝑣))
14 eqcom 2772 . . . . . . . . . . 11 ((𝐹𝑢) = (𝐹𝑣) ↔ (𝐹𝑣) = (𝐹𝑢))
1513, 14sylnib 319 . . . . . . . . . 10 ((𝐹𝑢) ⊊ (𝐹𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢))
1611, 15syl6 35 . . . . . . . . 9 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝑣 ∈ ω ∧ 𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
1716expdimp 444 . . . . . . . 8 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ 𝑣 ∈ ω) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1817adantrr 708 . . . . . . 7 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢𝑣 → ¬ (𝐹𝑣) = (𝐹𝑢)))
1910, 18jaod 885 . . . . . 6 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣𝑢𝑢𝑣) → ¬ (𝐹𝑣) = (𝐹𝑢)))
2019con2d 131 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → ¬ (𝑣𝑢𝑢𝑣)))
21 nnord 7271 . . . . . . 7 (𝑣 ∈ ω → Ord 𝑣)
22 nnord 7271 . . . . . . 7 (𝑢 ∈ ω → Ord 𝑢)
23 ordtri3 5944 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2421, 22, 23syl2an 589 . . . . . 6 ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2524adantl 473 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣𝑢𝑢𝑣)))
2620, 25sylibrd 250 . . . 4 (((𝑥 ≠ ∅ ∧ 𝑥 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
2726ralrimivva 3118 . . 3 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢))
28 frfnom 7734 . . . . . 6 (rec(𝐺, ∅) ↾ ω) Fn ω
29 fneq1 6157 . . . . . 6 (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
3028, 29mpbiri 249 . . . . 5 (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31 fvelrnb 6432 . . . . . . . 8 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹𝑣) = 𝑢))
32 inf3lem.4 . . . . . . . . . . . 12 𝐵 ∈ V
331, 2, 4, 32inf3lemd 8739 . . . . . . . . . . 11 (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥)
34 fvex 6388 . . . . . . . . . . . 12 (𝐹𝑣) ∈ V
3534elpw 4321 . . . . . . . . . . 11 ((𝐹𝑣) ∈ 𝒫 𝑥 ↔ (𝐹𝑣) ⊆ 𝑥)
3633, 35sylibr 225 . . . . . . . . . 10 (𝑣 ∈ ω → (𝐹𝑣) ∈ 𝒫 𝑥)
37 eleq1 2832 . . . . . . . . . 10 ((𝐹𝑣) = 𝑢 → ((𝐹𝑣) ∈ 𝒫 𝑥𝑢 ∈ 𝒫 𝑥))
3836, 37syl5ibcom 236 . . . . . . . . 9 (𝑣 ∈ ω → ((𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥))
3938rexlimiv 3174 . . . . . . . 8 (∃𝑣 ∈ ω (𝐹𝑣) = 𝑢𝑢 ∈ 𝒫 𝑥)
4031, 39syl6bi 244 . . . . . . 7 (𝐹 Fn ω → (𝑢 ∈ ran 𝐹𝑢 ∈ 𝒫 𝑥))
4140ssrdv 3767 . . . . . 6 (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
4241ancli 544 . . . . 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 6072 . . . 4 (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
4543, 44mpbir 222 . . 3 𝐹:ω⟶𝒫 𝑥
4627, 45jctil 515 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
47 dff13 6704 . 2 (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹𝑣) = (𝐹𝑢) → 𝑣 = 𝑢)))
4846, 47sylibr 225 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cin 3731  wss 3732  wpss 3733  c0 4079  𝒫 cpw 4315   cuni 4594  cmpt 4888  ran crn 5278  cres 5279  Ord word 5907   Fn wfn 6063  wf 6064  1-1wf1 6065  cfv 6068  ωcom 7263  reccrdg 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710
This theorem is referenced by:  inf3lem7  8746  dominf  9520  dominfac  9648
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