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Theorem phialllem2 4617
 Description: Lemma for phiall 4618. Any set without 0c is equal to the Phi of a set. (Contributed by Scott Fenton, 8-Apr-2021.)
Hypothesis
Ref Expression
phiall.1 A V
Assertion
Ref Expression
phialllem2 (¬ 0c Ax A = Phi x)
Distinct variable group:   x,A

Proof of Theorem phialllem2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 inss2 3476 . . 3 (ANn ) Nn
2 inss1 3475 . . . . 5 (ANn ) A
32sseli 3269 . . . 4 (0c (ANn ) → 0c A)
43con3i 127 . . 3 (¬ 0c A → ¬ 0c (ANn ))
5 phiall.1 . . . . 5 A V
6 nncex 4396 . . . . 5 Nn V
75, 6inex 4105 . . . 4 (ANn ) V
87phialllem1 4616 . . 3 (((ANn ) Nn ¬ 0c (ANn )) → y(ANn ) = Phi y)
91, 4, 8sylancr 644 . 2 (¬ 0c Ay(ANn ) = Phi y)
10 uncom 3408 . . . . . . 7 ((A Nn ) ∪ (ANn )) = ((ANn ) ∪ (A Nn ))
11 inundif 3628 . . . . . . 7 ((ANn ) ∪ (A Nn )) = A
1210, 11eqtri 2373 . . . . . 6 ((A Nn ) ∪ (ANn )) = A
13 uneq2 3412 . . . . . 6 ((ANn ) = Phi y → ((A Nn ) ∪ (ANn )) = ((A Nn ) ∪ Phi y))
1412, 13syl5eqr 2399 . . . . 5 ((ANn ) = Phi yA = ((A Nn ) ∪ Phi y))
15 phiun 4614 . . . . . 6 Phi ((A Nn ) ∪ y) = ( Phi (A Nn ) ∪ Phi y)
16 incom 3448 . . . . . . . . 9 ((A Nn ) ∩ Nn ) = ( Nn ∩ (A Nn ))
17 disjdif 3622 . . . . . . . . 9 ( Nn ∩ (A Nn )) =
1816, 17eqtri 2373 . . . . . . . 8 ((A Nn ) ∩ Nn ) =
19 phidisjnn 4615 . . . . . . . 8 (((A Nn ) ∩ Nn ) = Phi (A Nn ) = (A Nn ))
2018, 19ax-mp 5 . . . . . . 7 Phi (A Nn ) = (A Nn )
2120uneq1i 3414 . . . . . 6 ( Phi (A Nn ) ∪ Phi y) = ((A Nn ) ∪ Phi y)
2215, 21eqtri 2373 . . . . 5 Phi ((A Nn ) ∪ y) = ((A Nn ) ∪ Phi y)
2314, 22syl6eqr 2403 . . . 4 ((ANn ) = Phi yA = Phi ((A Nn ) ∪ y))
245, 6difex 4107 . . . . . 6 (A Nn ) V
25 vex 2862 . . . . . 6 y V
2624, 25unex 4106 . . . . 5 ((A Nn ) ∪ y) V
27 phieq 4570 . . . . . 6 (x = ((A Nn ) ∪ y) → Phi x = Phi ((A Nn ) ∪ y))
2827eqeq2d 2364 . . . . 5 (x = ((A Nn ) ∪ y) → (A = Phi xA = Phi ((A Nn ) ∪ y)))
2926, 28spcev 2946 . . . 4 (A = Phi ((A Nn ) ∪ y) → x A = Phi x)
3023, 29syl 15 . . 3 ((ANn ) = Phi yx A = Phi x)
3130exlimiv 1634 . 2 (y(ANn ) = Phi yx A = Phi x)
329, 31syl 15 1 (¬ 0c Ax A = Phi x)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550   Nn cnnc 4373  0cc0c 4374   Phi cphi 4562 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565 This theorem is referenced by:  phiall  4618
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