Theorem oawordriexmid 6614

Description: A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6613. (Contributed by Jim Kingdon, 15-May-2022.)

Hypothesis

Ref	Expression
oawordriexmid.1	⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))

Assertion

Ref	Expression
oawordriexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable groups:   𝑎,𝑏,𝑐   𝜑,𝑎

Allowed substitution hints:   𝜑(𝑏,𝑐)

Proof of Theorem oawordriexmid

Step	Hyp	Ref	Expression
1		1on 6567	. . . . 5 ⊢ 1o ∈ On
2		oawordriexmid.1	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
3	2	3expa 1227	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
4	3	expcom 116	. . . . . 6 ⊢ (𝑐 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))))
5	4	rgen 2583	. . . . 5 ⊢ ∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))
6		oveq2 6008	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑎 +o 𝑐) = (𝑎 +o 1o))
7		oveq2 6008	. . . . . . . . 9 ⊢ (𝑐 = 1o → (𝑏 +o 𝑐) = (𝑏 +o 1o))
8	6, 7	sseq12d 3255	. . . . . . . 8 ⊢ (𝑐 = 1o → ((𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐) ↔ (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))
9	8	imbi2d 230	. . . . . . 7 ⊢ (𝑐 = 1o → ((𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)) ↔ (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o))))
10	9	imbi2d 230	. . . . . 6 ⊢ (𝑐 = 1o → (((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) ↔ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))))
11	10	rspcv 2903	. . . . 5 ⊢ (1o ∈ On → (∀𝑐 ∈ On ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐))) → ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))))
12	1, 5, 11	mp2 16	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 1o) ⊆ (𝑏 +o 1o)))
13		oa1suc 6611	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 +o 1o) = suc 𝑎)
14	13	adantr 276	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 1o) = suc 𝑎)
15		oa1suc 6611	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑏 +o 1o) = suc 𝑏)
16	15	adantl 277	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +o 1o) = suc 𝑏)
17	14, 16	sseq12d 3255	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑎 +o 1o) ⊆ (𝑏 +o 1o) ↔ suc 𝑎 ⊆ suc 𝑏))
18	12, 17	sylibd 149	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏))
19	18	rgen2a 2584	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ⊆ 𝑏 → suc 𝑎 ⊆ suc 𝑏)
20	19	onsucsssucexmid 4618	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  Oncon0 4453  suc csuc 4455  (class class class)co 6000  1_oc1o 6553   +_o coa 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564

This theorem is referenced by: (None)