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Theorem nodenselem6 31570
Description: The restriction of a surreal to the abstraction from nodenselem4 31568 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem6
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nodenselem4 31568 . . . 4 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
21adantrl 751 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
3 nofnbday 31527 . . . . . 6 (𝐴 No 𝐴 Fn ( bday 𝐴))
43ad2antrr 761 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 Fn ( bday 𝐴))
5 nodenselem5 31569 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
6 bdayelon 31564 . . . . . . 7 ( bday 𝐴) ∈ On
76onelssi 5797 . . . . . 6 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ ( bday 𝐴))
85, 7syl 17 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ ( bday 𝐴))
9 fnssres 5964 . . . . 5 ((𝐴 Fn ( bday 𝐴) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ ( bday 𝐴)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) Fn {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
104, 8, 9syl2anc 692 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) Fn {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
11 resss 5383 . . . . . 6 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ 𝐴
12 rnss 5316 . . . . . 6 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ 𝐴 → ran (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ ran 𝐴)
1311, 12ax-mp 5 . . . . 5 ran (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ ran 𝐴
14 norn 31526 . . . . . 6 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
1514ad2antrr 761 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ran 𝐴 ⊆ {1𝑜, 2𝑜})
1613, 15syl5ss 3595 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ran (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ {1𝑜, 2𝑜})
17 df-f 5853 . . . 4 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}): {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}⟶{1𝑜, 2𝑜} ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) Fn {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∧ ran (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ⊆ {1𝑜, 2𝑜}))
1810, 16, 17sylanbrc 697 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}): {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}⟶{1𝑜, 2𝑜})
19 feq2 5986 . . . 4 (𝑥 = {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}):𝑥⟶{1𝑜, 2𝑜} ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}): {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}⟶{1𝑜, 2𝑜}))
2019rspcev 3295 . . 3 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}): {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}):𝑥⟶{1𝑜, 2𝑜})
212, 18, 20syl2anc 692 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ On (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}):𝑥⟶{1𝑜, 2𝑜})
22 elno 31521 . 2 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No ↔ ∃𝑥 ∈ On (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}):𝑥⟶{1𝑜, 2𝑜})
2321, 22sylibr 224 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  wss 3556  {cpr 4152   cint 4442   class class class wbr 4615  ran crn 5077  cres 5078  Oncon0 5684   Fn wfn 5844  wf 5845  cfv 5849  1𝑜c1o 7501  2𝑜c2o 7502   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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