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Theorem inf3lemd 9571
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9579 for detailed description. (Contributed by NM, 28-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
inf3lemd (𝐴 ∈ ω → (𝐹𝐴) ⊆ 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 inf3lemd
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6846 . . . . 5 (𝐴 = ∅ → (𝐹𝐴) = (𝐹‘∅))
2 inf3lem.1 . . . . . 6 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
3 inf3lem.2 . . . . . 6 𝐹 = (rec(𝐺, ∅) ↾ ω)
4 inf3lem.3 . . . . . 6 𝐴 ∈ V
5 inf3lem.4 . . . . . 6 𝐵 ∈ V
62, 3, 4, 5inf3lemb 9569 . . . . 5 (𝐹‘∅) = ∅
71, 6eqtrdi 2789 . . . 4 (𝐴 = ∅ → (𝐹𝐴) = ∅)
8 0ss 4360 . . . 4 ∅ ⊆ 𝑥
97, 8eqsstrdi 4002 . . 3 (𝐴 = ∅ → (𝐹𝐴) ⊆ 𝑥)
109a1d 25 . 2 (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
11 nnsuc 7824 . . . 4 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣)
12 vex 3451 . . . . . . . . . 10 𝑣 ∈ V
132, 3, 12, 5inf3lemc 9570 . . . . . . . . 9 (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹𝑣)))
1413eleq2d 2820 . . . . . . . 8 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹𝑣))))
15 vex 3451 . . . . . . . . . 10 𝑢 ∈ V
16 fvex 6859 . . . . . . . . . 10 (𝐹𝑣) ∈ V
172, 3, 15, 16inf3lema 9568 . . . . . . . . 9 (𝑢 ∈ (𝐺‘(𝐹𝑣)) ↔ (𝑢𝑥 ∧ (𝑢𝑥) ⊆ (𝐹𝑣)))
1817simplbi 499 . . . . . . . 8 (𝑢 ∈ (𝐺‘(𝐹𝑣)) → 𝑢𝑥)
1914, 18syl6bi 253 . . . . . . 7 (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢𝑥))
2019ssrdv 3954 . . . . . 6 (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥)
21 fveq2 6846 . . . . . . 7 (𝐴 = suc 𝑣 → (𝐹𝐴) = (𝐹‘suc 𝑣))
2221sseq1d 3979 . . . . . 6 (𝐴 = suc 𝑣 → ((𝐹𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥))
2320, 22syl5ibrcom 247 . . . . 5 (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹𝐴) ⊆ 𝑥))
2423rexlimiv 3142 . . . 4 (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹𝐴) ⊆ 𝑥)
2511, 24syl 17 . . 3 ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹𝐴) ⊆ 𝑥)
2625expcom 415 . 2 (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥))
2710, 26pm2.61ine 3025 1 (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wrex 3070  {crab 3406  Vcvv 3447  cin 3913  wss 3914  c0 4286  cmpt 5192  cres 5639  suc csuc 6323  cfv 6500  ωcom 7806  reccrdg 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360
This theorem is referenced by:  inf3lem2  9573  inf3lem3  9574  inf3lem6  9577
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