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Theorem nosepdmlem 27713
Description: Lemma for nosepdm 27714. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
nosepdmlem ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepdmlem
StepHypRef Expression
1 sltval2 27686 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)})))
2 fvex 6914 . . . . . . 7 (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ∈ V
3 fvex 6914 . . . . . . 7 (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ∈ V
42, 3brtp 5529 . . . . . 6 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ↔ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
5 df-3or 1085 . . . . . . . . . 10 ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ↔ ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
6 ndmfv 6936 . . . . . . . . . . . . 13 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
7 1oex 8506 . . . . . . . . . . . . . . . . . . 19 1o ∈ V
87prid1 4771 . . . . . . . . . . . . . . . . . 18 1o ∈ {1o, 2o}
98nosgnn0i 27689 . . . . . . . . . . . . . . . . 17 ∅ ≠ 1o
10 neeq1 2993 . . . . . . . . . . . . . . . . 17 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 1o ↔ ∅ ≠ 1o))
119, 10mpbiri 257 . . . . . . . . . . . . . . . 16 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 1o)
1211neneqd 2935 . . . . . . . . . . . . . . 15 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ¬ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o)
1312intnanrd 488 . . . . . . . . . . . . . 14 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ¬ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅))
1412intnanrd 488 . . . . . . . . . . . . . 14 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ¬ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o))
15 ioran 981 . . . . . . . . . . . . . 14 (¬ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ↔ (¬ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∧ ¬ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
1613, 14, 15sylanbrc 581 . . . . . . . . . . . . 13 ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ¬ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
176, 16syl 17 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 → ¬ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
1817adantl 480 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ) ∧ ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴) → ¬ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
19 orel1 886 . . . . . . . . . . 11 (¬ (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → (((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
2018, 19syl 17 . . . . . . . . . 10 (((𝐴 No 𝐵 No ) ∧ ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴) → (((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
215, 20biimtrid 241 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴) → ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)))
22 ndmfv 6936 . . . . . . . . . . . . 13 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅)
23 2on 8510 . . . . . . . . . . . . . . . . 17 2o ∈ On
2423elexi 3484 . . . . . . . . . . . . . . . 16 2o ∈ V
2524prid2 4772 . . . . . . . . . . . . . . 15 2o ∈ {1o, 2o}
2625nosgnn0i 27689 . . . . . . . . . . . . . 14 ∅ ≠ 2o
27 neeq1 2993 . . . . . . . . . . . . . 14 ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 2o ↔ ∅ ≠ 2o))
2826, 27mpbiri 257 . . . . . . . . . . . . 13 ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 2o)
2922, 28syl 17 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ 2o)
3029neneqd 2935 . . . . . . . . . . 11 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵 → ¬ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)
3130con4i 114 . . . . . . . . . 10 ((𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)
3231adantl 480 . . . . . . . . 9 (((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)
3321, 32syl6 35 . . . . . . . 8 (((𝐴 No 𝐵 No ) ∧ ¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴) → ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3433ex 411 . . . . . . 7 ((𝐴 No 𝐵 No ) → (¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 → ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)))
3534com23 86 . . . . . 6 ((𝐴 No 𝐵 No ) → ((((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o) ∨ ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = ∅ ∧ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 2o)) → (¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)))
364, 35biimtrid 241 . . . . 5 ((𝐴 No 𝐵 No ) → ((𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)))
371, 36sylbid 239 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → (¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵)))
38373impia 1114 . . 3 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → (¬ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
3938orrd 861 . 2 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
40 elun 4148 . 2 ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵) ↔ ( {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ dom 𝐵))
4139, 40sylibr 233 1 ((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845  w3o 1083  w3a 1084   = wceq 1534  wcel 2099  wne 2930  {crab 3419  cun 3945  c0 4325  {ctp 4637  cop 4639   cint 4954   class class class wbr 5153  dom cdm 5682  Oncon0 6376  cfv 6554  1oc1o 8489  2oc2o 8490   No csur 27669   <s cslt 27670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-dm 5692  df-ord 6379  df-on 6380  df-suc 6382  df-iota 6506  df-fv 6562  df-1o 8496  df-2o 8497  df-slt 27673
This theorem is referenced by:  nosepdm  27714
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