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Theorem sltintdifex 27032
Description: If 𝐴 <s 𝐵, then the intersection of all the ordinals that have differing signs in 𝐴 and 𝐵 exists. (Contributed by Scott Fenton, 22-Feb-2012.)
Assertion
Ref Expression
sltintdifex ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem sltintdifex
StepHypRef Expression
1 sltval2 27027 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
2 fvex 6859 . . . 4 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
3 fvex 6859 . . . 4 (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
42, 3brtp 5484 . . 3 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
5 fvprc 6838 . . . . . . 7 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
6 1n0 8438 . . . . . . . . 9 1o ≠ ∅
76neii 2942 . . . . . . . 8 ¬ 1o = ∅
8 eqeq1 2737 . . . . . . . . 9 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ↔ ∅ = 1o))
9 eqcom 2740 . . . . . . . . 9 (∅ = 1o ↔ 1o = ∅)
108, 9bitrdi 287 . . . . . . . 8 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ↔ 1o = ∅))
117, 10mtbiri 327 . . . . . . 7 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o)
125, 11syl 17 . . . . . 6 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o)
1312con4i 114 . . . . 5 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
1413adantr 482 . . . 4 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
1513adantr 482 . . . 4 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
16 fvprc 6838 . . . . . . 7 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
17 2on0 8432 . . . . . . . . 9 2o ≠ ∅
1817neii 2942 . . . . . . . 8 ¬ 2o = ∅
19 eqeq1 2737 . . . . . . . . 9 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o ↔ ∅ = 2o))
20 eqcom 2740 . . . . . . . . 9 (∅ = 2o ↔ 2o = ∅)
2119, 20bitrdi 287 . . . . . . . 8 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o ↔ 2o = ∅))
2218, 21mtbiri 327 . . . . . . 7 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)
2316, 22syl 17 . . . . . 6 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)
2423con4i 114 . . . . 5 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
2524adantl 483 . . . 4 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
2614, 15, 253jaoi 1428 . . 3 ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
274, 26sylbi 216 . 2 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V)
281, 27syl6bi 253 1 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3o 1087   = wceq 1542  wcel 2107  wne 2940  {crab 3406  Vcvv 3447  c0 4286  {ctp 4594  cop 4596   cint 4911   class class class wbr 5109  Oncon0 6321  cfv 6500  1oc1o 8409  2oc2o 8410   No csur 27011   <s cslt 27012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fv 6508  df-1o 8416  df-2o 8417  df-slt 27015
This theorem is referenced by:  sltres  27033
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