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Theorem karden 9890
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 10546). 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 9889 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 5152 . . . . . . 7 (𝑀 = 𝐴 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝐴 β‰ˆ 𝐴))
31enref 8981 . . . . . . 7 𝐴 β‰ˆ 𝐴
41, 2, 3ceqsexv2d 3529 . . . . . 6 βˆƒπ‘€ 𝑀 β‰ˆ 𝐴
5 abn0 4381 . . . . . 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 9881 . . . . . 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 2994 . . . . 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 4386 . . . 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 3479 . . . . . . . 8 𝑧 ∈ V
13 breq1 5152 . . . . . . . 8 (𝑀 = 𝑧 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝑧 β‰ˆ 𝐴))
1412, 13elab 3669 . . . . . . 7 (𝑧 ∈ {𝑀 ∣ 𝑀 β‰ˆ 𝐴} ↔ 𝑧 β‰ˆ 𝐴)
15 breq1 5152 . . . . . . . 8 (𝑀 = 𝑦 β†’ (𝑀 β‰ˆ 𝐴 ↔ 𝑦 β‰ˆ 𝐴))
1615ralab 3688 . . . . . . 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 5152 . . . . . . . . . . . . 13 (π‘₯ = 𝑧 β†’ (π‘₯ β‰ˆ 𝐴 ↔ 𝑧 β‰ˆ 𝐴))
26 fveq2 6892 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘§))
2726sseq1d 4014 . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . 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 8999 . . . . . . . 8 (𝑧 β‰ˆ 𝐴 β†’ 𝐴 β‰ˆ 𝑧)
42 entr 9002 . . . . . . . 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 3154 . . 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 9118 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (π‘₯ β‰ˆ 𝐴 ↔ π‘₯ β‰ˆ 𝐡))
50 enen2 9118 . . . . . . 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 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544   β‰ˆ cen 8936  rankcrnk 9758
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-r1 9759  df-rank 9760
This theorem is referenced by: (None)
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