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Theorem karden 9935
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 10591). 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 9934 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)
Hypotheses
Ref Expression
karden.a 𝐴 ∈ V
karden.c 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
karden.d 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Assertion
Ref Expression
karden (𝐶 = 𝐷𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem karden
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 karden.a . . . . . . 7 𝐴 ∈ V
2 breq1 5146 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
31enref 9025 . . . . . . 7 𝐴𝐴
41, 2, 3ceqsexv2d 3533 . . . . . 6 𝑤 𝑤𝐴
5 abn0 4385 . . . . . 6 ({𝑤𝑤𝐴} ≠ ∅ ↔ ∃𝑤 𝑤𝐴)
64, 5mpbir 231 . . . . 5 {𝑤𝑤𝐴} ≠ ∅
7 scott0 9926 . . . . . 6 ({𝑤𝑤𝐴} = ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
87necon3bii 2993 . . . . 5 ({𝑤𝑤𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
96, 8mpbi 230 . . . 4 {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
10 rabn0 4389 . . . 4 ({𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦))
119, 10mpbi 230 . . 3 𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
12 vex 3484 . . . . . . . 8 𝑧 ∈ V
13 breq1 5146 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
1412, 13elab 3679 . . . . . . 7 (𝑧 ∈ {𝑤𝑤𝐴} ↔ 𝑧𝐴)
15 breq1 5146 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
1615ralab 3697 . . . . . . 7 (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
1714, 16anbi12i 628 . . . . . 6 ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18 simpl 482 . . . . . . . . 9 ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴)
1918a1i 11 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴))
20 karden.c . . . . . . . . . . . 12 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21 karden.d . . . . . . . . . . . 12 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2220, 21eqeq12i 2755 . . . . . . . . . . 11 (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
23 abbib 2811 . . . . . . . . . . 11 ({𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
2422, 23bitri 275 . . . . . . . . . 10 (𝐶 = 𝐷 ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25 breq1 5146 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
26 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
2726sseq1d 4015 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
2827imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
2928albidv 1920 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3025, 29anbi12d 632 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31 breq1 5146 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
3227imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3332albidv 1920 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3431, 33anbi12d 632 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3530, 34bibi12d 345 . . . . . . . . . . 11 (𝑥 = 𝑧 → (((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))))
3635spvv 1996 . . . . . . . . . 10 (∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3724, 36sylbi 217 . . . . . . . . 9 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
38 simpl 482 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵)
3937, 38biimtrdi 253 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵))
4019, 39jcad 512 . . . . . . 7 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧𝐴𝑧𝐵)))
41 ensym 9043 . . . . . . . 8 (𝑧𝐴𝐴𝑧)
42 entr 9046 . . . . . . . 8 ((𝐴𝑧𝑧𝐵) → 𝐴𝐵)
4341, 42sylan 580 . . . . . . 7 ((𝑧𝐴𝑧𝐵) → 𝐴𝐵)
4440, 43syl6 35 . . . . . 6 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴𝐵))
4517, 44biimtrid 242 . . . . 5 (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴𝐵))
4645expd 415 . . . 4 (𝐶 = 𝐷 → (𝑧 ∈ {𝑤𝑤𝐴} → (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵)))
4746rexlimdv 3153 . . 3 (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵))
4811, 47mpi 20 . 2 (𝐶 = 𝐷𝐴𝐵)
49 enen2 9158 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
50 enen2 9158 . . . . . . 7 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
5150imbi1d 341 . . . . . 6 (𝐴𝐵 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5251albidv 1920 . . . . 5 (𝐴𝐵 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5349, 52anbi12d 632 . . . 4 (𝐴𝐵 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
5453abbidv 2808 . . 3 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
5554, 20, 213eqtr4g 2802 . 2 (𝐴𝐵𝐶 = 𝐷)
5648, 55impbii 209 1 (𝐶 = 𝐷𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  cfv 6561  cen 8982  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-r1 9804  df-rank 9805
This theorem is referenced by: (None)
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