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Theorem elsetrecslem 42215
Description: Lemma for elsetrecs 42216. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 42214. To see why this lemma also requires setrec1 42209, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
elsetrecs.1 𝐵 = setrecs(𝐹)
Assertion
Ref Expression
elsetrecslem (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem elsetrecslem
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ssdifsn 4316 . . . . 5 (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵𝐵 ∧ ¬ 𝐴𝐵))
21simprbi 480 . . . 4 (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴𝐵)
32con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4 elsetrecs.1 . . . 4 𝐵 = setrecs(𝐹)
5 sseq1 3624 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐵𝑎𝐵))
6 fveq2 6189 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
76eleq2d 2686 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴 ∈ (𝐹𝑎)))
85, 7anbi12d 747 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
98notbid 308 . . . . . . 7 (𝑥 = 𝑎 → (¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
109spv 2259 . . . . . 6 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
11 imnan 438 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
12 idd 24 . . . . . . . . . . 11 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ¬ 𝐴 ∈ (𝐹𝑎)))
13 vex 3201 . . . . . . . . . . . . 13 𝑎 ∈ V
1413a1i 11 . . . . . . . . . . . 12 (𝑎𝐵𝑎 ∈ V)
15 id 22 . . . . . . . . . . . 12 (𝑎𝐵𝑎𝐵)
164, 14, 15setrec1 42209 . . . . . . . . . . 11 (𝑎𝐵 → (𝐹𝑎) ⊆ 𝐵)
1712, 16jctild 566 . . . . . . . . . 10 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1817a2i 14 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1911, 18sylbir 225 . . . . . . . 8 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
2019adantrd 484 . . . . . . 7 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → ((𝑎𝐵 ∧ ¬ 𝐴𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
21 ssdifsn 4316 . . . . . . 7 (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎𝐵 ∧ ¬ 𝐴𝑎))
22 ssdifsn 4316 . . . . . . 7 ((𝐹𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎)))
2320, 21, 223imtr4g 285 . . . . . 6 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2410, 23syl 17 . . . . 5 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2524alrimiv 1854 . . . 4 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
264, 25setrec2v 42214 . . 3 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
273, 26nsyl 135 . 2 (𝐴𝐵 → ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
28 df-ex 1704 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
2927, 28sylibr 224 1 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1480   = wceq 1482  wex 1703  wcel 1989  Vcvv 3198  cdif 3569  wss 3572  {csn 4175  cfv 5886  setrecscsetrecs 42201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-r1 8624  df-rank 8625  df-setrecs 42202
This theorem is referenced by:  elsetrecs  42216
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