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Theorem karden 8796
 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 9411). 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 8795 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.)
Hypotheses
Ref Expression
karden.1 𝐴 ∈ V
karden.2 𝐵 ∈ V
karden.3 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
karden.4 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.1 . . . . . . . 8 𝐴 ∈ V
21enref 8030 . . . . . . 7 𝐴𝐴
3 breq1 4688 . . . . . . . 8 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
41, 3spcev 3331 . . . . . . 7 (𝐴𝐴 → ∃𝑤 𝑤𝐴)
52, 4ax-mp 5 . . . . . 6 𝑤 𝑤𝐴
6 abn0 3987 . . . . . 6 ({𝑤𝑤𝐴} ≠ ∅ ↔ ∃𝑤 𝑤𝐴)
75, 6mpbir 221 . . . . 5 {𝑤𝑤𝐴} ≠ ∅
8 scott0 8787 . . . . . 6 ({𝑤𝑤𝐴} = ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
98necon3bii 2875 . . . . 5 ({𝑤𝑤𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
107, 9mpbi 220 . . . 4 {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
11 rabn0 3991 . . . 4 ({𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦))
1210, 11mpbi 220 . . 3 𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
13 vex 3234 . . . . . . . 8 𝑧 ∈ V
14 breq1 4688 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
1513, 14elab 3382 . . . . . . 7 (𝑧 ∈ {𝑤𝑤𝐴} ↔ 𝑧𝐴)
16 breq1 4688 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
1716ralab 3400 . . . . . . 7 (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
1815, 17anbi12i 733 . . . . . 6 ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
19 simpl 472 . . . . . . . . 9 ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴)
2019a1i 11 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴))
21 karden.3 . . . . . . . . . . . 12 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
22 karden.4 . . . . . . . . . . . 12 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2321, 22eqeq12i 2665 . . . . . . . . . . 11 (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
24 abbi 2766 . . . . . . . . . . 11 (∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
2523, 24bitr4i 267 . . . . . . . . . 10 (𝐶 = 𝐷 ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
26 breq1 4688 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
27 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
2827sseq1d 3665 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
2928imbi2d 329 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3029albidv 1889 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3126, 30anbi12d 747 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
32 breq1 4688 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
3328imbi2d 329 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3433albidv 1889 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3532, 34anbi12d 747 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3631, 35bibi12d 334 . . . . . . . . . . 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))))))
3736spv 2296 . . . . . . . . . 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)))))
3825, 37sylbi 207 . . . . . . . . 9 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
39 simpl 472 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵)
4038, 39syl6bi 243 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵))
4120, 40jcad 554 . . . . . . 7 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧𝐴𝑧𝐵)))
42 ensym 8046 . . . . . . . 8 (𝑧𝐴𝐴𝑧)
43 entr 8049 . . . . . . . 8 ((𝐴𝑧𝑧𝐵) → 𝐴𝐵)
4442, 43sylan 487 . . . . . . 7 ((𝑧𝐴𝑧𝐵) → 𝐴𝐵)
4541, 44syl6 35 . . . . . 6 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴𝐵))
4618, 45syl5bi 232 . . . . 5 (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴𝐵))
4746expd 451 . . . 4 (𝐶 = 𝐷 → (𝑧 ∈ {𝑤𝑤𝐴} → (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵)))
4847rexlimdv 3059 . . 3 (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵))
4912, 48mpi 20 . 2 (𝐶 = 𝐷𝐴𝐵)
50 enen2 8142 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
51 enen2 8142 . . . . . . 7 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
5251imbi1d 330 . . . . . 6 (𝐴𝐵 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5352albidv 1889 . . . . 5 (𝐴𝐵 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5450, 53anbi12d 747 . . . 4 (𝐴𝐵 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
5554abbidv 2770 . . 3 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
5655, 21, 223eqtr4g 2710 . 2 (𝐴𝐵𝐶 = 𝐷)
5749, 56impbii 199 1 (𝐶 = 𝐷𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948   class class class wbr 4685  ‘cfv 5926   ≈ cen 7994  rankcrnk 8664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-r1 8665  df-rank 8666 This theorem is referenced by: (None)
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