Theorem r1ord3 3501

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478.

Assertion

Ref	Expression
r1ord3	⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B → (R1 ‘A) ⊆ (R1 ‘B)))

Proof of Theorem r1ord3

Step	Hyp	Ref	Expression
1		onsseleq 2254	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B ↔ (A ∈ B ∨ A = B)))
2		r1ord2 3500	. . . 4 ⊢ (B ∈ On → (A ∈ B → (R1 ‘A) ⊆ (R1 ‘B)))
3	2	adantl 305	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → (R1 ‘A) ⊆ (R1 ‘B)))
4		fveq2 2832	. . . . 5 ⊢ (A = B → (R1 ‘A) = (R1 ‘B))
5		eqimss 1548	. . . . 5 ⊢ ((R1 ‘A) = (R1 ‘B) → (R1 ‘A) ⊆ (R1 ‘B))
6	4, 5	syl 12	. . . 4 ⊢ (A = B → (R1 ‘A) ⊆ (R1 ‘B))
7	6	a1i 7	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (A = B → (R1 ‘A) ⊆ (R1 ‘B)))
8	3, 7	jaod 329	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → ((A ∈ B ∨ A = B) → (R1 ‘A) ⊆ (R1 ‘B)))
9	1, 8	sylbid 178	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B → (R1 ‘A) ⊆ (R1 ‘B)))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Oncon0 2199   ‘cfv 2422  R₁cr1 3485

This theorem is referenced by:  r1val1 3502  rankr1lem 3517  ssrankr1 3520  rankel 3524  rankval3 3525  bndrank 3526  r1pwcl 3530  rankr1id 3539  ranklon 3540

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-r1 3487

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