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Theorem nodenselem7 31571
Description: Lemma for nodense 31573. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
nodenselem7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝐶,𝑎

Proof of Theorem nodenselem7
StepHypRef Expression
1 nodenselem4 31568 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
21adantrl 751 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
3 onelon 5709 . . . . 5 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝐶 ∈ On)
43ex 450 . . . 4 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → 𝐶 ∈ On))
52, 4syl 17 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → 𝐶 ∈ On))
62, 3sylan 488 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝐶 ∈ On)
7 ontri1 5718 . . . . . . . . . . . . . 14 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶 ↔ ¬ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
87con2bid 344 . . . . . . . . . . . . 13 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
98biimpd 219 . . . . . . . . . . . 12 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
109ex 450 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
1110com23 86 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
122, 11syl 17 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
1312imp 445 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
146, 13mpd 15 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)
15 intss1 4459 . . . . . . 7 (𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)
1614, 15nsyl 135 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
1716ex 450 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
18 fveq2 6150 . . . . . . . 8 (𝑎 = 𝐶 → (𝐴𝑎) = (𝐴𝐶))
19 fveq2 6150 . . . . . . . 8 (𝑎 = 𝐶 → (𝐵𝑎) = (𝐵𝐶))
2018, 19neeq12d 2851 . . . . . . 7 (𝑎 = 𝐶 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝐶) ≠ (𝐵𝐶)))
2120elrab 3347 . . . . . 6 (𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2221notbii 310 . . . . 5 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2317, 22syl6ib 241 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶))))
24 imnan 438 . . . 4 ((𝐶 ∈ On → ¬ (𝐴𝐶) ≠ (𝐵𝐶)) ↔ ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2523, 24syl6ibr 242 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ (𝐴𝐶) ≠ (𝐵𝐶))))
265, 25mpdd 43 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐴𝐶) ≠ (𝐵𝐶)))
27 df-ne 2791 . . 3 ((𝐴𝐶) ≠ (𝐵𝐶) ↔ ¬ (𝐴𝐶) = (𝐵𝐶))
2827con2bii 347 . 2 ((𝐴𝐶) = (𝐵𝐶) ↔ ¬ (𝐴𝐶) ≠ (𝐵𝐶))
2926, 28syl6ibr 242 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  wss 3556   cint 4442   class class class wbr 4615  Oncon0 5684  cfv 5849   No csur 31515   <s cslt 31516   bday cbday 31517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-1o 7508  df-2o 7509  df-no 31518  df-slt 31519  df-bday 31520
This theorem is referenced by:  nodense  31573
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