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Theorem elsetrecslem 48316
Description: Lemma for elsetrecs 48317. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 48313. To see why this lemma also requires setrec1 48308, 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 4793 . . . . 5 (𝐵 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝐵𝐵 ∧ ¬ 𝐴𝐵))
21simprbi 495 . . . 4 (𝐵 ⊆ (𝐵 ∖ {𝐴}) → ¬ 𝐴𝐵)
32con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐵 ⊆ (𝐵 ∖ {𝐴}))
4 elsetrecs.1 . . . 4 𝐵 = setrecs(𝐹)
5 sseq1 4002 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥𝐵𝑎𝐵))
6 fveq2 6896 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
76eleq2d 2811 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴 ∈ (𝐹𝑎)))
85, 7anbi12d 630 . . . . . . . 8 (𝑥 = 𝑎 → ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
98notbid 317 . . . . . . 7 (𝑥 = 𝑎 → (¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎))))
109spvv 1992 . . . . . 6 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
11 imnan 398 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) ↔ ¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)))
12 idd 24 . . . . . . . . . . 11 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ¬ 𝐴 ∈ (𝐹𝑎)))
13 vex 3465 . . . . . . . . . . . . 13 𝑎 ∈ V
1413a1i 11 . . . . . . . . . . . 12 (𝑎𝐵𝑎 ∈ V)
15 id 22 . . . . . . . . . . . 12 (𝑎𝐵𝑎𝐵)
164, 14, 15setrec1 48308 . . . . . . . . . . 11 (𝑎𝐵 → (𝐹𝑎) ⊆ 𝐵)
1712, 16jctild 524 . . . . . . . . . 10 (𝑎𝐵 → (¬ 𝐴 ∈ (𝐹𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1817a2i 14 . . . . . . . . 9 ((𝑎𝐵 → ¬ 𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
1911, 18sylbir 234 . . . . . . . 8 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎𝐵 → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
2019adantrd 490 . . . . . . 7 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → ((𝑎𝐵 ∧ ¬ 𝐴𝑎) → ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎))))
21 ssdifsn 4793 . . . . . . 7 (𝑎 ⊆ (𝐵 ∖ {𝐴}) ↔ (𝑎𝐵 ∧ ¬ 𝐴𝑎))
22 ssdifsn 4793 . . . . . . 7 ((𝐹𝑎) ⊆ (𝐵 ∖ {𝐴}) ↔ ((𝐹𝑎) ⊆ 𝐵 ∧ ¬ 𝐴 ∈ (𝐹𝑎)))
2320, 21, 223imtr4g 295 . . . . . 6 (¬ (𝑎𝐵𝐴 ∈ (𝐹𝑎)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2410, 23syl 17 . . . . 5 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → (𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
2524alrimiv 1922 . . . 4 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → ∀𝑎(𝑎 ⊆ (𝐵 ∖ {𝐴}) → (𝐹𝑎) ⊆ (𝐵 ∖ {𝐴})))
264, 25setrec2v 48313 . . 3 (∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐵 ⊆ (𝐵 ∖ {𝐴}))
273, 26nsyl 140 . 2 (𝐴𝐵 → ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
28 df-ex 1774 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) ↔ ¬ ∀𝑥 ¬ (𝑥𝐵𝐴 ∈ (𝐹𝑥)))
2927, 28sylibr 233 1 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  cdif 3941  wss 3944  {csn 4630  cfv 6549  setrecscsetrecs 48300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9617  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-rank 9790  df-setrecs 48301
This theorem is referenced by:  elsetrecs  48317
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