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Theorem karden 9836
Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10492). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9835 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {π‘₯ ∣ π‘₯ β‰ˆ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)
Hypotheses
Ref Expression
karden.a 𝐴 ∈ V
karden.c 𝐢 = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
karden.d 𝐷 = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
Assertion
Ref Expression
karden (𝐢 = 𝐷 ↔ 𝐴 β‰ˆ 𝐡)
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐡,𝑦
Allowed substitution hints:   𝐢(π‘₯,𝑦)   𝐷(π‘₯,𝑦)

Proof of Theorem karden
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 karden.a . . . . . . 7 𝐴 ∈ V
2 breq1 5109 . . . . . . 7 (𝑀 = 𝐴 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ 𝐴))
31enref 8928 . . . . . . 7 𝐴 β‰ˆ 𝐴
41, 2, 3ceqsexv2d 3496 . . . . . 6 βˆƒπ‘€ 𝑀 β‰ˆ 𝐴
5 abn0 4341 . . . . . 6 ({𝑀 ∣ 𝑀 β‰ˆ 𝐴} β‰  βˆ… ↔ βˆƒπ‘€ 𝑀 β‰ˆ 𝐴)
64, 5mpbir 230 . . . . 5 {𝑀 ∣ 𝑀 β‰ˆ 𝐴} β‰  βˆ…
7 scott0 9827 . . . . . 6 ({𝑀 ∣ 𝑀 β‰ˆ 𝐴} = βˆ… ↔ {𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∣ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)} = βˆ…)
87necon3bii 2993 . . . . 5 ({𝑀 ∣ 𝑀 β‰ˆ 𝐴} β‰  βˆ… ↔ {𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∣ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)} β‰  βˆ…)
96, 8mpbi 229 . . . 4 {𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∣ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)} β‰  βˆ…
10 rabn0 4346 . . . 4 ({𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∣ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)} β‰  βˆ… ↔ βˆƒπ‘§ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴}βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))
119, 10mpbi 229 . . 3 βˆƒπ‘§ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴}βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)
12 vex 3448 . . . . . . . 8 𝑧 ∈ V
13 breq1 5109 . . . . . . . 8 (𝑀 = 𝑧 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝑧 β‰ˆ 𝐴))
1412, 13elab 3631 . . . . . . 7 (𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ↔ 𝑧 β‰ˆ 𝐴)
15 breq1 5109 . . . . . . . 8 (𝑀 = 𝑦 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝑦 β‰ˆ 𝐴))
1615ralab 3650 . . . . . . 7 (βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦) ↔ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))
1714, 16anbi12i 628 . . . . . 6 ((𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∧ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)) ↔ (𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))
18 simpl 484 . . . . . . . . 9 ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ 𝑧 β‰ˆ 𝐴)
1918a1i 11 . . . . . . . 8 (𝐢 = 𝐷 β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ 𝑧 β‰ˆ 𝐴))
20 karden.c . . . . . . . . . . . 12 𝐢 = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
21 karden.d . . . . . . . . . . . 12 𝐷 = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
2220, 21eqeq12i 2751 . . . . . . . . . . 11 (𝐢 = 𝐷 ↔ {π‘₯ ∣ (π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))})
23 abbib 2805 . . . . . . . . . . 11 ({π‘₯ ∣ (π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} ↔ βˆ€π‘₯((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))))
2422, 23bitri 275 . . . . . . . . . 10 (𝐢 = 𝐷 ↔ βˆ€π‘₯((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))))
25 breq1 5109 . . . . . . . . . . . . 13 (π‘₯ = 𝑧 β†’ (π‘₯ β‰ˆ 𝐴 ↔ 𝑧 β‰ˆ 𝐴))
26 fveq2 6843 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘§))
2726sseq1d 3976 . . . . . . . . . . . . . . 15 (π‘₯ = 𝑧 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))
2827imbi2d 341 . . . . . . . . . . . . . 14 (π‘₯ = 𝑧 β†’ ((𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ (𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))
2928albidv 1924 . . . . . . . . . . . . 13 (π‘₯ = 𝑧 β†’ (βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))
3025, 29anbi12d 632 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ ((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))))
31 breq1 5109 . . . . . . . . . . . . 13 (π‘₯ = 𝑧 β†’ (π‘₯ β‰ˆ 𝐡 ↔ 𝑧 β‰ˆ 𝐡))
3227imbi2d 341 . . . . . . . . . . . . . 14 (π‘₯ = 𝑧 β†’ ((𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ (𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))
3332albidv 1924 . . . . . . . . . . . . 13 (π‘₯ = 𝑧 β†’ (βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))
3431, 33anbi12d 632 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ ((π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))))
3530, 34bibi12d 346 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ (((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))) ↔ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))))))
3635spvv 2001 . . . . . . . . . 10 (βˆ€π‘₯((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))) β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))))
3724, 36sylbi 216 . . . . . . . . 9 (𝐢 = 𝐷 β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)))))
38 simpl 484 . . . . . . . . 9 ((𝑧 β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ 𝑧 β‰ˆ 𝐡)
3937, 38syl6bi 253 . . . . . . . 8 (𝐢 = 𝐷 β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ 𝑧 β‰ˆ 𝐡))
4019, 39jcad 514 . . . . . . 7 (𝐢 = 𝐷 β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ (𝑧 β‰ˆ 𝐴 ∧ 𝑧 β‰ˆ 𝐡)))
41 ensym 8946 . . . . . . . 8 (𝑧 β‰ˆ 𝐴 β†’ 𝐴 β‰ˆ 𝑧)
42 entr 8949 . . . . . . . 8 ((𝐴 β‰ˆ 𝑧 ∧ 𝑧 β‰ˆ 𝐡) β†’ 𝐴 β‰ˆ 𝐡)
4341, 42sylan 581 . . . . . . 7 ((𝑧 β‰ˆ 𝐴 ∧ 𝑧 β‰ˆ 𝐡) β†’ 𝐴 β‰ˆ 𝐡)
4440, 43syl6 35 . . . . . 6 (𝐢 = 𝐷 β†’ ((𝑧 β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦))) β†’ 𝐴 β‰ˆ 𝐡))
4517, 44biimtrid 241 . . . . 5 (𝐢 = 𝐷 β†’ ((𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ∧ βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦)) β†’ 𝐴 β‰ˆ 𝐡))
4645expd 417 . . . 4 (𝐢 = 𝐷 β†’ (𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} β†’ (βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦) β†’ 𝐴 β‰ˆ 𝐡)))
4746rexlimdv 3147 . . 3 (𝐢 = 𝐷 β†’ (βˆƒπ‘§ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴}βˆ€π‘¦ ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘¦) β†’ 𝐴 β‰ˆ 𝐡))
4811, 47mpi 20 . 2 (𝐢 = 𝐷 β†’ 𝐴 β‰ˆ 𝐡)
49 enen2 9065 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (π‘₯ β‰ˆ 𝐴 ↔ π‘₯ β‰ˆ 𝐡))
50 enen2 9065 . . . . . . 7 (𝐴 β‰ˆ 𝐡 β†’ (𝑦 β‰ˆ 𝐴 ↔ 𝑦 β‰ˆ 𝐡))
5150imbi1d 342 . . . . . 6 (𝐴 β‰ˆ 𝐡 β†’ ((𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ (𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))))
5251albidv 1924 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))))
5349, 52anbi12d 632 . . . 4 (𝐴 β‰ˆ 𝐡 β†’ ((π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))))
5453abbidv 2802 . . 3 (𝐴 β‰ˆ 𝐡 β†’ {π‘₯ ∣ (π‘₯ β‰ˆ 𝐴 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} = {π‘₯ ∣ (π‘₯ β‰ˆ 𝐡 ∧ βˆ€π‘¦(𝑦 β‰ˆ 𝐡 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))})
5554, 20, 213eqtr4g 2798 . 2 (𝐴 β‰ˆ 𝐡 β†’ 𝐢 = 𝐷)
5648, 55impbii 208 1 (𝐢 = 𝐷 ↔ 𝐴 β‰ˆ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283   class class class wbr 5106  β€˜cfv 6497   β‰ˆ cen 8883  rankcrnk 9704
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-r1 9705  df-rank 9706
This theorem is referenced by: (None)
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